Baseband equivalent volterra series for digital predistortion in multi-band power amplifiers

ABSTRACT

Methods, systems and apparatus for modelling a power amplifier and pre-distorter fed by a multi-band signal are disclosed. According to one aspect, a method includes receiving a multi-band signal and generating a discrete base band equivalent, BBE, Volterra series based on the received multi-band signal, where the series has distortion products grouped according to determined shared kernels. The shared kernels are determined based on a transformation of a real-valued continuous-time pass band Volterra series without pruning of kernels.

TECHNICAL FIELD

The present invention relates to amplifiers and transmitters, and inparticular to a method and system for digital pre-distortion inmulti-band power amplifiers and transmitters.

BACKGROUND

Advanced modulation techniques and access technologies are enabling highspeed mobile access for users. However, these techniques are increasingthe complexity of the development of radio transceivers. The continuedquest for flexible and dynamic networks challenges designers to developnovel radio systems capable of processing multi-band and frequencyaggregated multi-standard, multi-carrier communication signals. Whileradio systems designers could use multiple power amplifiers (PA) 10, 12,each one dedicated to a particular radio frequency (RF) band, as shownin FIG. 1, this solution dramatically increases the deployment cost ofthe network and limits network flexibility. Alternatively, a moresuitable solution for future communication systems is the use of aunique multi-band PA 14 to amplify combined multi-band multi-carrier andmulti-standard signals, as shown in FIG. 2. This would incur lower costsfor materials and more flexibility in deployment. However, this solutionimposes new efficiency and linearity challenges. In fact, a singlemulti-band PA should provide RF performance (efficiency, gain, outputpower) comparable to multiple single-band PA modules. In addition, whenconcurrently driven with multiple signals scattered over spacedfrequencies, a multi-band PA can actually aggravate the distortionproblems encountered.

Previous efforts to improve the efficiency and linearity of single-bandPAs, such as load (Doherty) and drain-supply (envelope tracking)modulations, have been applied to improve efficiency at the back-offregion of single-band PAs. Recent studies identified sources ofbandwidth limitations and devised solutions to mitigate them. Severalproof-of-concept prototypes have demonstrated excellent efficiency inthe back-off region over a wide range of frequencies.

On the other hand, linearization techniques, such as DigitalPre-distortion (DPD), have been applied to extend the linear region ofsingle-band PAs. A number of DPD schemes have been developed which havedemonstrated excellent linearization capability. These schemes evolvedfrom low complexity schemes (e.g., memory-less polynomials, Hammersteinand Wiener models, memory polynomials) to more comprehensive ones (e.g.,Volterra series and Artificial Neural Networks (ANN)).

In the case of the Volterra series, its application to the linearizationof single-band PAs which exhibit significant memory effects wasconditional on its successful pruning. This motivated researchersinvestigating multi-band DPD schemes to discard the Volterra seriesoption, worrying it would lead to unmanageable and impracticalsolutions. Hence, most of the recent work has concentrated on efforts togeneralize the previously mentioned low complexity schemes to thedual-band PA context.

A dual-band signal can be expressed as follows:

$\begin{matrix}\begin{matrix}{{x(t)} = {{x_{1}(t)} + {x_{2}(t)}}} \\{{= {{Re}\left( {{{{\overset{\sim}{x}}_{1}(t)}{\mathbb{e}}^{{j\omega}_{1}t}} + {{{\overset{\sim}{x}}_{2}(t)}{\mathbb{e}}^{{j\omega}_{2}t}}} \right)}},}\end{matrix} & (1)\end{matrix}$where x(t) is the combined dual-band dual-standard signal, x₁(t) andx₂(t) are single-band multicarrier signals modulated around the angularfrequencies ω₁ and ω₂, respectively, and {tilde over (x)}₁(t) {tildeover (x)}₂(t) denote the baseband envelops of x₁(t) and x₂(t),respectively.

The dual-band input signal can be represented as a broadband signal withan angular carrier frequency equal to (ω₁+ω₂)/2 as given by:

$\begin{matrix}\begin{matrix}{{x(t)} = {{x_{1}(t)} + {x_{2}(t)}}} \\{= {{Re}\left( {\left( {{{{\overset{\sim}{x}}_{1}(t)}{\mathbb{e}}^{j\frac{\omega_{1} - \omega_{2}}{2}t}} + {{{\overset{\sim}{x}}_{2}(t)}{\mathbb{e}}^{j\frac{\omega_{2} - \omega_{1}}{2}t}}} \right){\mathbb{e}}^{j\frac{\omega_{1} + \omega_{2}}{2}t}} \right)}} \\{{= {{Re}\left( {{\overset{\sim}{x}(t)} \cdot {\mathbb{e}}^{j\frac{\omega_{1} + \omega_{2}}{2}t}} \right)}},}\end{matrix} & (2)\end{matrix}$where {tilde over (x)}(t) is the baseband envelope of the combinedsignal. When the dual-band signal is amplified by a PA, the passbandcomponent of the output signal, y_(pb)(t), can be described as:

$\begin{matrix}\begin{matrix}{{y_{pb}(t)} = {{y_{1}(t)} + {y_{2}(t)}}} \\{= {{Re}\left( {\left( {{{{\overset{\sim}{y}}_{1}(t)}{\mathbb{e}}^{j\frac{\omega_{1} - \omega_{2}}{2}t}} + {{{\overset{\sim}{y}}_{2}(t)}{\mathbb{e}}^{j\frac{\omega_{2} - \omega_{1}}{2}t}}} \right) \cdot {\mathbb{e}}^{j\frac{\omega_{1} + \omega_{2}}{2}t}} \right)}} \\{{= {{Re}\left( {{{\overset{\sim}{y}}_{pb}(t)} \cdot {\mathbb{e}}^{j\frac{\omega_{1} + \omega_{2}}{2}t}} \right)}},}\end{matrix} & (3)\end{matrix}$where y₁(t) and y₂(t) are multicarrier output signals modulated aroundthe angular frequencies ω₁ and ω₂ respectively, and {tilde over (y)}₁(t){tilde over (y)}₂(t) denote the baseband envelopes of y₁(t) and y₂(t),respectively.

In the classical PA behavioral modeling approach, the PA behavior ismodeled as a single-input single-output (SISO) system where the PAoutput {tilde over (y)}_(pb)(t) is a function of the PA input {tildeover (x)}(t), as given in (4):{tilde over (y)} _(pb)(t)={tilde over (f)}({tilde over (x)}(t))  (4)where {tilde over (f)} is the SISO describing function of the PA 16, asshown in FIG. 3. Note that the output shown in FIG. 3 is idealized.Digitization of the SISO model requires sampling both {tilde over(x)}(t) and {tilde over (y)}_(pb)(t) at a high frequency rate asfollows:

${f_{s,{SiSo}} \geq \left( {S + {5 \cdot {\max\left( {\frac{B_{1}}{2},\frac{B_{2}}{2}} \right)}}} \right)},$where B₁ and B₂ represent the bandwidths of {tilde over (x)}₁(t) and{tilde over (x)}₂(t), respectively, and S denotes the frequency spacingbetween the two signals

$\left( {{i.e.},{= {{f_{2} - f_{1}} = \frac{\omega_{2} - \omega_{1}}{2\pi}}}} \right),$and where f₁ and f₂ are the two bands' carrier frequencies,respectively. The factor of 5 represents the spectrum regrowth due to PAnonlinearity which is assumed equal to 5.

Alternatively, a dual-input dual-output (DIDO) approach would require asignificantly lower sampling rate. In such a formulation, the PA outputin each band (i.e.,{tilde over (y)}₁(t) and {tilde over (y)}₂(t), isexpressed separately as a function of the two input signals' envelopes{tilde over (x)}₁(t) and {tilde over (x)}₂(t), as given by:{tilde over (y)} ₁(t)={tilde over (f)} ₁({tilde over (x)} ₁(t), {tildeover (x)} ₂(t)){tilde over (y)} ₂(t)={tilde over (f)} ₂({tilde over (x)} ₁(t), {tildeover (x)} ₂(t))   (5)where {tilde over (f)}₁ and {tilde over (f)}₂ form the PA's 18 dual-banddescribing functions, as shown in FIG. 4. Note that the output shown inFIG. 4 is idealized. Actual output depends on the success of thepre-distortion approach employed. The construction of the two describingfunctions, {tilde over (g)}₁ and {tilde over (g)}₂, needed to modeland/or to linearize the dual-band PA, is performed in the digital domainThis requires the sampling of {tilde over (x)}₁(t), {tilde over(x)}₂(t), {tilde over (y)}₁(t) and {tilde over (y)}₂(t) at a frequencyrate given byf _(s,DiDo)≧(5·max(B ₁ ,B ₂))

This sampling rate is independent of the frequency separation, S, whichmay be very large. Hence, f_(s,DiDo) is significantly lower thanf_(s,SiSo). For example, if we assume a dual-band signal composed of a15 MHz WCDMA signal around 2.1 GHz and a 10 MHz LTE one centered at 2.4GHz, the theoretical sampling frequency needed for the dual-band model,f_(s,DiDo), has to be at least equal to 75 MHz; significantly lower thanthe 675 MHz sampling frequency required for the SISO model. The ratiobetween the two sampling frequencies is equal to

$\frac{f_{s,{SiSo}}}{f_{s,{DiDo}}} = {0.11.}$

There have been several attempts to devise describing functions in orderto implement a dual-band model as given in equation (5). Some haveproposed a third order frequency selective pre-distortion technique tohandle each band separately in order to model and/or linearize PAsexhibiting strong “differential” memory effects (i.e., high imbalancebetween the upper and lower in-band and inter-band distortioncomponents). This technique was tested using a multi-carrier 1001 WCDMAsignal and extended to address the 5^(th) order inter-modulationdistortions of a PA driven with multi-tone signals. Although thistechnique was applied to multicarrier single-band signals, it can begeneralized to the dual-band case provided the required sampling rate isreduced to cope with large frequency spacing.

Some have proposed an IF dual-band model implementing aWeiner-Hammerstein DPD scheme using a sub-sampling feedback path.Although the reported simulation results showed 10 dB spectrum regrowthreduction, the proposed architecture involved digital to analogconversion (DAC) and analog to digital conversion (ADC) withdisproportionate sampling rates and complicated IF processing.Furthermore, starting with a 5^(th) order memoryless model driven with adual-band signal, some have shown that the PA's output in each banddepends on both PA input signals. This observation has been generalizedto the memory polynomial model to yield a two dimension DPD (2D-DPD)model. Reported linearization results demonstrated a 12 dB improvementof the adjacent channel leakage ratio (ACLR) at the cost of a largenumber of coefficients. However, stability issues were reported.

Some have proposed an orthogonal representation to handle theill-conditioning problem and numerical instability of the 2D-DPD model.Alternatively, some have proposed 2D Hammerstein and 2D Weiner models toaddress the large number of coefficients required by the 2D-DPD model.When applied to construct a behavioral model of a dual-band PA with anonlinearity order equal to 5 and a memory depth equal to 5, the 2DHammerstein and 2D Weiner models needed 40 coefficients in each band asopposed to the 2D-DPD which required 150 coefficients. However, whilethe 2D-DPD model has been validated as a dual-band digitalpre-distorter, the application of the 2D Hammerstein and 2D Weinermodels to the linearization of dual PAs is problematic and onlybehavioral modeling results have been reported.

Some have pointed out the implementation complexity of the 2D-DPD andhave suggested a two dimensional look up table (LUT)-basedrepresentation as an alternative. This latter approach was furthersimplified to use single dimension LUTs. When applied to thelinearization of a dual-band PA driven with dual-band signals (separatedby 97 MHz), the model demonstrated an ACLR of about −45 dB, which barelypasses the mask. However, the proposed DPD scheme was operated with asampling rate equal to 153.6 MHz and consequently a large oversamplingrate with a 10 MHz signal. Hardware to achieve such a large oversamplingrate is costly and undesirable.

Known behavior modeling and linearization approaches have beenrestricted to generalizing low complexity schemes for single-band PAs.Volterra series have been avoided due to the perceived unmanageablenumber of coefficients and consequent complexity.

SUMMARY

Methods, systems and apparatus for modelling a power amplifier andpre-distorter fed by a multi-band signal are disclosed. According to oneaspect, a method includes receiving a multi-band signal and generating adiscrete base band equivalent, BBE, Volterra series based on thereceived multi-band signal, where the series has distortion productsgrouped according to determined shared kernels. The shared kernels aredetermined based on a transformation of a real-valued continuous-timepass band Volterra series without pruning of kernels.

According to this aspect, in some embodiments, the shared kernels aredetermined based on the transformation of the real-valuedcontinuous-time pass band Volterra series by steps that includetransforming the real-valued continuous time pass band Volterra seriesto a multi-frequency complex-valued envelope series. The multi-frequencycomplex-valued envelope signal is transformed to a continuous-time passband-only series, which is then transformed to a continuous-timebaseband equivalent series. The continuous-time baseband equivalentsignal is discretized to produce the discrete base band equivalentVolterra series. Shared kernels are identify, each shared kernel havingdistortion products in common with another shared kernel. In someembodiments, transforming the continuous-time pass band-only signal to acontinuous-time baseband equivalent signal includes expressing thecontinuous-time pass band-only series in convolution form. The Laplacetransform is applied to the convolution form to produce a Laplace domainexpression, which is frequency shifted to baseband to produce a basebandequivalent expression in the Laplace domain The inverse Laplacetransform is applied to the baseband equivalent expression to producethe continuous-time baseband equivalent series. In some embodiments, anumber of terms in the Laplace domain expression are reduced viasymmetry. In some embodiments, terms of the Laplace domain expressionare grouped based on frequency intervals where distortion terms are notzero. In some embodiments, discretizing the continuous-time basebandequivalent series to produce the discrete base band equivalent Volterraseries includes truncating the continuous-time baseband equivalentseries to a finite non-linearity order, and expressing the truncatedseries as summations of non-linear distortion terms, with upper limitsof the summations being memory depths assigned to each order of thenon-linear distortion terms. In some embodiments, a distortion term is agroup of distortion products multiplied by a shared kernel.

According to another aspect, embodiments include a digital pre-distorter(DPD) system. The system includes a Volterra series DPD modelling unit.The DPD modelling unit is configured to calculate a discrete base bandequivalent, BBE, Volterra series. The series has distortion productsgrouped according to determined shared kernels. The shared kernels aredetermined based on a transformation of a real-valued continuous-timepass band Volterra series without pruning of kernels.

According to this aspect, the DPD may further comprise a power amplifierconfigured to produce an output in response to a multi-band input. Theoutput of the power amplifier is provided to the Volterra series DPDmodelling unit to enable the Volterra series DPD modeling unit tocompute the shared kernels based on the output of the power amplifier.In some embodiments, the DPD system further comprises a transmitterobservation receiver configured to sample the output of the poweramplifier and provide the sampled output to the Volterra series DPDmodelling unit. In some embodiments, the distortion products and theirassociated kernels are determined by transforming the real-valuedcontinuous time pass band Volterra series to a discrete base bandequivalent Volterra series according to a series of steps that include:transforming the real-valued continuous time pass band Volterra seriesto a multi-frequency complex-valued envelope series; transforming themulti-frequency complex-valued envelope signal to a continuous-time passband-only series; transforming the continuous-time pass band-only signalto a continuous-time baseband equivalent series; discretizing thecontinuous-time baseband equivalent signal to produce the discrete baseband equivalent Volterra series. The shared kernels are identified suchthat each shared kernel has distortion products in common with anothershared kernel. In some embodiments, transforming the continuous-timepass band-only signal to a continuous-time baseband equivalent signalincludes the following steps: expressing the continuous-time passband-only series in convolution form; applying a Laplace transform tothe convolution form to produce a Laplace domain expression; frequencyshifting the Laplace domain expression to baseband to produce a basebandequivalent expression in the Laplace domain; and applying an inverseLaplace transform to the baseband equivalent expression to produce thecontinuous-time baseband equivalent series. In some embodiments, anumber of terms in the Laplace domain expression are reduced viasymmetry. In some embodiments, transforming the continuous-time passband-only signal to a continuous-time baseband equivalent signal furtherincludes grouping terms of the Laplace domain expression based onfrequency intervals where distortion terms are not zero. In someembodiments, discretizing the continuous-time baseband equivalent seriesto produce the discrete base band equivalent Volterra series includes:truncating the continuous-time baseband equivalent series to a finitenon-linearity order; and expressing the truncated series as summationsof non-linear distortion terms, with upper limits of the summationsbeing memory depths assigned to each order of the non-linear distortionterms. In some embodiments, a distortion term is a group of distortionproducts multiplied by a shared kernel.

According to another aspect, embodiments include a Volterra seriesdigital pre-distorter, DPD, modelling unit. The DPD modelling unitincludes a memory module configured to store terms of a discrete baseband equivalent, BBE, Volterra series. Also, a grouping module isconfigured to group distortion products of the series according todetermined shared kernels. The DPD modelling unit also includes a sharedkernel determiner configured to determine the shared kernels based on atransformation of a real-valued continuous-time pass band Volterraseries without pruning of kernels. Also, a series term calculator isconfigured to calculate the terms of the discrete base band equivalentVolterra series, the terms being the distortion products multiplied bytheir respective shared kernels.

According to this aspect, in some embodiments, the BBE Volterra seriesterms are based on a multi-band input. In some embodiments, themulti-band input is a dual band input. In some embodiments, the sharedkernel determiner is further configured to determine the shared kernelsvia a least squares estimate based on the multi-band input and an outputof a power amplifier. In some embodiments, the kernels and distortionproducts are derived from the real-valued continuous-time pass bandVolterra series by: transforming the real-valued continuous time passband Volterra series to a multi-frequency complex-valued envelopeseries; transforming the multi-frequency complex-valued envelope signalto a continuous-time pass band-only series; transforming thecontinuous-time pass band-only signal to a continuous-time basebandequivalent series; discretizing the continuous-time baseband equivalentsignal to produce the discrete base band equivalent Volterra series; andidentifying the shared kernels, each shared kernel having distortionproducts in common

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram of a known power amplification architectureusing a separate power amplifier per input signal;

FIG. 2 is a block diagram of a known power amplification architectureusing a single power amplifier;

FIG. 3 is an idealized signal input/output diagram for a known singleinput/single output system;

FIG. 4 is an idealized signal input/output diagram for a known dualinput/dual output system;

FIG. 5 is a block diagram of grouping of distortion terms groupedaccording to whether the terms are self-distortion terms or inter-banddistortion terms;

FIG. 6 is a plot of outputs of a power amplifier with no pre-distortion,with 2D DPD, and with dual band base band equivalent (BBE) Volterraseries DPD for a first input signal of a first dual band input signal;

FIG. 7 is a plot of outputs of a power amplifier with no pre-distortion,with 2D DPD, and with dual band base band equivalent (BBE) Volterraseries DPD for a second input signals of the first dual band inputsignal;

FIG. 8 is a plot of outputs of a power amplifier with no pre-distortion,with 2D DPD, and with dual band base band equivalent (BBE) Volterraseries DPD for a first input signal of a second dual band input signal;

FIG. 9 is a plot of outputs of a power amplifier with no pre-distortion,with 2D DPD, and with dual band base band equivalent (BBE) Volterraseries DPD for a second input signals of the second dual band inputsignal;

FIG. 10 is a plot of outputs of a power amplifier with nopre-distortion, with 2D DPD, and with dual band base band equivalent(BBE) Volterra series DPD for a first input signal of a third dual bandinput signal;

FIG. 11 is a plot of outputs of a power amplifier with nopre-distortion, with 2D DPD, and with dual band base band equivalent(BBE) Volterra series DPD for a second input signals of the third dualband input signal;

FIG. 12 is a block diagram of a digital pre-distortion poweramplification system constructed in accordance with principles of thepresent invention;

FIG. 13 is a block diagram of a DPD modelling unit constructed inaccordance with principles of the present invention;

FIG. 14 is a flowchart of an exemplary process of modelling a poweramplifier fed by a multi-band input signal;

FIG. 15 is a flowchart of an exemplary process of transforming areal-valued continuous-time pass band Volterra series to a discrete baseband equivalent (BBE) Volterra series;

FIG. 16 is a flowchart of an exemplary process of transforming acontinuous-time pass band series to a continuous-time basebandequivalent series; and

FIG. 17 is a flowchart of an exemplary process of discretizing thecontinuous-time baseband equivalent series to produce the discrete baseband equivalent Volterra series.

DETAILED DESCRIPTION

Before describing in detail exemplary embodiments that are in accordancewith the present invention, it is noted that the embodiments resideprimarily in combinations of apparatus components and processing stepsrelated to digital pre-distortion of wideband power amplifiers fed by amulti-band signal. Accordingly, the system and method components havebeen represented where appropriate by conventional symbols in thedrawings, showing only those specific details that are pertinent tounderstanding the embodiments of the present invention so as not toobscure the disclosure with details that will be readily apparent tothose of ordinary skill in the art having the benefit of the descriptionherein.

As used herein, relational terms, such as “first” and “second,” “top”and “bottom,” and the like, may be used solely to distinguish one entityor element from another entity or element without necessarily requiringor implying any physical or logical relationship or order between suchentities or elements.

The Volterra series is an appropriate modeling framework for dual-bandPAs which are recursive nonlinear dynamic systems with fading memory.The empirically pruned LPE Volterra series has been successfully appliedto model and to linearize single-band PAs. However, application of aVolterra series to model and linearize a multi-band PA without pruningand having a manageable number of terms has not been presented. In someembodiments described herein, a BBE dual-band Volterra seriesformulation is derived from the original passband real-valued Volterraseries to model and linearize dual-band PAs. This approachadvantageously does not require pruning and the derivation set forthbelow is particularly attentive to addressing exponential growth in thenumber of coefficients experienced with the LPE approach. Thus, thepresent arrangement provides a method and system for modelling a poweramplifier fed by a multi-band signal input. Methods described hereinpresent a formulation of pre-distortion that transforms acontinuous-time real valued Volterra series to a discrete base bandequivalent Volterra series that has a reduced number of terms and isachieved without pruning. Steps for deriving the model are describedbelow.

Step 1: Continuous-time real-valued Volterra series modeling: TheVolterra series framework is initially used to describe the relationshipbetween the real pass-band signals at the system input and output:y(t)=Σ_(p=1) ^(NL)∫_(−∞) ^(∞)∫_(−∞) ^(∞) h _(p)(τ₁, . . . ,τ_(p))Π_(j=1) ^(p) x(t−τ _(j))dτ _(j)  (6)where x(t) and y(t) represent the PA input and output RF signals and NLis the nonlinearity order.

Step 2: Real-valued to complex-valued envelope signal transformation: Inthe case of a dual-band PA, the band-limited input signal x(t) can beexpressed as:

$\begin{matrix}{{x(t)} = {{{Re}\left\{ {{{{\overset{\sim}{x}}_{1}(t)}{\mathbb{e}}^{{j\omega}_{1}t}} + {{{\overset{\sim}{x}}_{2}(t)}{\mathbb{e}}^{{j\omega}_{2}t}}} \right\}} = {{\frac{1}{2}\left( {{{{\overset{\sim}{x}}_{1}^{*}(t)}{\mathbb{e}}^{{- {j\omega}_{1}}t}} + {{{\overset{\sim}{x}}_{1}(t)}{\mathbb{e}}^{{j\omega}_{1}t}}} \right)} +}}} & (7) \\{\mspace{79mu}{\frac{1}{2}\left( {{{{\overset{\sim}{x}}_{2}^{*}(t)}{\mathbb{e}}^{{- {j\omega}_{2}}t}} + {\overset{\sim}{x}}_{2{(t)}{\mathbb{e}}^{{j\omega}_{2}t}}} \right)}} & (8)\end{matrix}$where {tilde over (x)}₁(t) and {tilde over (x)}₂(t) represent the twocomplex baseband envelope signals that modulate the two differentangular frequencies, ω₁ and ω₂. Substituting (8) into (7) yields anexpression relating the output signal y(t) to {tilde over (x)}₁(t),{tilde over (x)}₂(t), ω₁ to ω₂ as follows:y(t)=ƒ({tilde over (x)} ₁(t),{tilde over (x)} ₂(t), e ^(±jpω) ¹ ^(t) , e^(±jqω) ² ^(t)); (p,q)ε{1, . . . NL} ²  (9)where the describing function ƒ is used to represent the real valuedVolterra series (7). Since the output signal y(t) in (9) is a result ofthe application of a nonlinear function to a band-limited RF signal, itcontains several spectrum components that involve multiple envelopes,{tilde over (y)}_(p,q)(t), which modulate the mixing products, pω₁±qω₂,of ω₁ and ω₂ as shown in (10):

$\begin{matrix}{{{y(t)} = {{\sum\limits_{p,{q = {NL}},{- {NL}}}^{{p + q} = {NL}}\;{\frac{1}{2}\left( {{{{\overset{\sim}{y}}_{p,q}^{*}(t)} \cdot {\mathbb{e}}^{{- {j{({{p\;\omega_{1}} + {q\;\omega_{2}}})}}}t}} + {{{\overset{\sim}{y}}_{p,q}(t)} \cdot {\mathbb{e}}^{{j{({{p\;\omega_{1}} + {q\;\omega_{2}}})}}t}}} \right)}} = \;{{\frac{1}{2}\left( {{{{\overset{\sim}{y}}_{0,0}^{*}(t)} \cdot {\mathbb{e}}^{{- 0}\; j\; t}} + {{{\overset{\sim}{y}}_{0,0}(t)} \cdot {\mathbb{e}}^{0j\; t}}} \right)} + {\frac{1}{2}\left( {{{\overset{\sim}{y}}_{1,0}^{*}(t)} \cdot {\mathbb{e}}^{{- j}\;\omega_{1}t}} \right)} + {\frac{1}{2}\left( {{{{\overset{\sim}{y}}_{1,0}^{*}(t)} \cdot {\mathbb{e}}^{{- j}\;\omega_{1}t}} + {{{\overset{\sim}{y}}_{1,0}(t)}{\mathbb{e}}^{j\;\omega_{1}t}}} \right)} + {\frac{1}{2}\left( {{{{\overset{\sim}{y}}_{0,1}^{*}(t)} \cdot {\mathbb{e}}^{{- j}\;\omega_{2}t}} + {{{\overset{\sim}{y}}_{0,1}(t)}{\mathbb{e}}^{j\;\omega_{2}t}}} \right)} + {\frac{1}{2}\left( {{{{\overset{\sim}{y}}_{2,{- 1}}^{*}(t)} \cdot {\mathbb{e}}^{{- {j{({{2\omega_{1}} - \omega_{2}})}}}t}} + {{{\overset{\sim}{y}}_{2,{- 1}}(t)}{\mathbb{e}}^{{j{({{2\omega_{1}} - \omega_{2}})}}t}}} \right)} + {\frac{1}{2}\left( {{{{\overset{\sim}{y}}_{{- 1},2}^{*}(t)} \cdot {\mathbb{e}}^{{- {j{({{2\omega_{2}} - \omega_{1}})}}}t}} + {{{\overset{\sim}{y}}_{{- 1},2}(t)}{\mathbb{e}}^{{j{({{2\omega_{2}} - \omega_{1}})}}t}}} \right)}}}}\mspace{11mu}\;{\vdots\mspace{14mu} + {\frac{1}{2}\left( {{{{\overset{\sim}{y}}_{{NL},0}^{*}(t)} \cdot {\mathbb{e}}^{{NL}\;{j\omega}_{1}t}} + {{{\overset{\sim}{y}}_{{NL},0}(t)}{\mathbb{e}}^{{NL}\;{j\omega}_{1}t}}} \right)} + {\frac{1}{2}\left( {{{{\overset{\sim}{y}}_{0,{NL}}^{*}(t)} \cdot {\mathbb{e}}^{{- {NL}}\;{j\omega}_{2}t}} + {{{\overset{\sim}{y}}_{0,{NL}}(t)}{\mathbb{e}}^{{NL}\;{j\omega}_{2}t}}} \right)}}} & (10)\end{matrix}$Here, {tilde over (y)}_0,0(t) denotes the envelope at DC, {tilde over(y)}_1,0(t) and {tilde over (y)}_0,1(t) denote the envelopes of thefirst order in-band signals, and {tilde over (y)}_(2, −1)(t) and {tildeover (y)}_(−1,2)(t), denote the envelopes of the third order inter-bandsignals. Finally, {tilde over (y)}_(0,NL)(t) and {tilde over(y)}_(NL,0)(t), represent the first and second NLth harmonics,respectively.

Step 3: Multi frequency to passband only transformation: Equating theterms on the right sides of (9) and (10) that share the same frequencyrange (fundamental, mixing products) yields a multi-frequency modelconsisting of several distinct equations that relate the outputenvelopes {tilde over (y)}_(p,q)(t) to {tilde over (x)}₁(t), {tilde over(x)}₂(t), ω₁ and ω₂. Since we are mainly interested in the relationshipbetween the envelopes of the output and input signals around the twocarriers' frequencies, only the passband components of the PA output areconsidered in the equation below:

$\begin{matrix}{{y_{pb}(t)} = {{{\frac{1}{2}\left( {{{{\overset{\sim}{y}}_{\omega_{1}}^{*}(t)} \cdot {\mathbb{e}}^{{- j}\;\omega_{1}t}} + {{{\overset{\sim}{y}}_{\omega_{1}}(t)}{\mathbb{e}}^{j\;\omega_{1}t}}} \right)} + {\frac{1}{2}\left( {{{{\overset{\sim}{y}}_{\omega_{2}}^{*}(t)} \cdot {\mathbb{e}}^{{- j}\;\omega_{2}t}} + {{{\overset{\sim}{y}}_{\omega_{2}}(t)}{\mathbb{e}}^{j\;\omega_{2}t}}} \right)}} = {{\frac{1}{2}\left( {y_{\omega_{1}}^{*}(t)} \right)} + {\frac{1}{2}\left( {{y_{\omega_{2}}^{*}(t)} + {y_{\omega_{2}}(t)}} \right)}}}} & (11)\end{matrix}$where y_(ω) ₁ (t)={tilde over (y)}_(ω) ₁ (t)e^(jω) ¹ ^(t) and y_(ω) ₂(t)={tilde over (y)}_(ω) ₂ (t)e^(jω) ² ^(t)

Additional derivations are applied to produce the detailed expressionfor the first term in (11), y_(ω) ₁ (t), around the first frequencySimilar derivations can be used to produce the expression of the secondfrequency term, y_(ω) ₂ (t), which can be modeled as a summation of theVolterra series nonlinear terms y_(ω) ₁ _(,2k+1)(t) of order 2k+1. Theterm can also be expressed as a function of the envelopes of thenonlinear terms, y_(ω) ₁ _(,2k+1)(t), denoted hereafter {tilde over(y)}_(ω) ₁ _(,2k+1)(t) and the angular frequency ω₁ as follows:

$\begin{matrix}{{y_{\omega_{1}}(t)} = {{\sum\limits_{k = 0}^{\infty}\;{y_{\omega_{1},{{2k} + 1}}(t)}} = {\left( {\sum\limits_{k = 0}^{\infty}\;{{\overset{\sim}{y}}_{\omega_{1},{{2k} + 1}}(t)}} \right) \cdot {\mathbb{e}}^{j\;\omega_{1}t}}}} & (12)\end{matrix}$

It is worth noting that only odd powered terms are retained and eventerms are discarded since they do not appear in the pass band response.Equating the terms on the right sides of expanded (9) and (12) yields acontinuous BBE DIDO Volterra series that expresses y_(ω) ₁ _(,2k+1)(t)as functions of {tilde over (x)}(t), and ω₁. Below is the expression fory_(ω) ₁ _(,1)(t) and y_(ω) ₁ _(,3)(t).

$\begin{matrix}{{{y_{\omega_{1},1}(t)} = {\int_{- \infty}^{\infty}{{{h_{1}\left( \tau_{1} \right)} \cdot {{\overset{\sim}{x}}_{1}\left( {t - \tau_{1}} \right)}}{{\mathbb{e}}^{j\;{\omega_{1}{({t - \tau_{1}})}}} \cdot \ {\mathbb{d}\tau_{1}}}}}}{y_{\omega_{1},3}(t)} = {{\int_{- \infty}^{\infty}{\int_{- \infty}^{\infty}{\int_{- \infty}^{\infty}\ {{{h_{3}\left( {\tau_{1},\tau_{2},\tau_{3}} \right)} \cdot \left( {{{\overset{\sim}{x}}_{1}\left( {t - \tau_{1}} \right)}{\mathbb{e}}^{j\;{\omega_{1}{({t - \tau_{1}})}}}} \right)}\left( {{{\overset{\sim}{x}}_{1}\left( {t - \tau_{2}} \right)}{\mathbb{e}}^{j\;{\omega_{1}{({t - \tau_{2}})}}}} \right)\left( {{\overset{\sim}{x}\left( {t - \tau_{3}} \right)}{\mathbb{e}}^{j\;{\omega_{1}{({t - \tau_{2}})}}}} \right)^{*}}}}} + {\left( {{{\overset{\sim}{x}}_{1}\left( {t - \tau_{1}} \right)}{\mathbb{e}}^{j\;{\omega_{1}{({t - \tau_{1}})}}}} \right)\left( {{{\overset{\sim}{x}}_{1}\left( {t - \tau_{2}} \right)}{\mathbb{e}}^{j\;{\omega_{1}{({t - \tau_{2}})}}}} \right)^{*}\left( {{{\overset{\sim}{x}}_{1}\left( {t - \tau_{3}} \right)}{\mathbb{e}}^{j\;{\omega_{1}{({t - \tau_{3}})}}}} \right)} + \mspace{50mu}{\left( {{{\overset{\sim}{x}}_{1}\left( {t - \tau_{1}} \right)}{\mathbb{e}}^{j\;{\omega_{1}{({t - \tau_{1}})}}}} \right)\left( {{{\overset{\sim}{x}}_{1}\left( {t - \tau_{2}} \right)}{\mathbb{e}}^{j\;{\omega_{1}{({t - \tau_{2}})}}}} \right)\left( {{{\overset{\sim}{x}}_{1}\left( {t - \tau_{3}} \right)}{\mathbb{e}}^{j\;{\omega_{2}{({t - \tau_{3}})}}}} \right)^{*}} + {\left( {{{\overset{\sim}{x}}_{1}\left( {t - \tau_{1}} \right)}{\mathbb{e}}^{j\;{\omega_{1}{({t - \tau_{1}})}}}} \right)\left( {{{\overset{\sim}{x}}_{2}\left( {t - \tau_{2}} \right)}{\mathbb{e}}^{j\;{\omega_{2}{({t - \tau_{2}})}}}} \right)^{*}} + {\left( {{{\overset{\sim}{x}}_{1}\left( {t - \tau_{1}} \right)}{\mathbb{e}}^{j\;{\omega_{1}{({t - \tau_{1}})}}}} \right)\left( {{{\overset{\sim}{x}}_{2}\left( {t - \tau_{2}} \right)}{\mathbb{e}}^{{j\omega}_{2}{({t - \tau_{2}})}}} \right)^{*}\left( {{{\overset{\sim}{x}}_{2}\left( {t - \tau_{3}} \right)}{\mathbb{e}}^{{j\omega}_{2}{({t - \tau_{3}})}}} \right)} + {\left( {{{\overset{\sim}{x}}_{2}\left( {t - \tau_{1}} \right)}{\mathbb{e}}^{j\;{\omega_{2}{({t - \tau_{1}})}}}} \right)^{*}\left( {{{\overset{\sim}{x}}_{1}\left( {t - \tau_{2}} \right)}{\mathbb{e}}^{j\;{\omega_{1}{({t - \tau_{2}})}}}} \right)\left( {{{\overset{\sim}{x}}_{2}\left( {t - \tau_{3}} \right)}{\mathbb{e}}^{j\;{\omega_{2}{({t - \tau_{3}})}}}} \right)} + {\left( {{{\overset{\sim}{x}}_{2}\left( {t - \tau_{1}} \right)}{\mathbb{e}}^{{j\omega}_{2}{({t - \tau_{1}})}}} \right)\left( {{{\overset{\sim}{x}}_{1}\left( {t - \tau_{2}} \right)}{\mathbb{e}}^{j\;{\omega_{1}{({t - \tau_{2}})}}}} \right)\left( {{{\overset{\sim}{x}}_{2}\left( {t - \tau_{3}} \right)}{\mathbb{e}}^{{j\omega}_{2}{({t - \tau_{3}})}}} \right)^{*}} + {\left( {{{\overset{\sim}{x}}_{2}\left( {t - \tau_{1}} \right)}{\mathbb{e}}^{{j\omega}_{2}{({t - \tau_{1}})}}} \right)\left( {{{\overset{\sim}{x}}_{2}\left( {t - \tau_{2}} \right)}{\mathbb{e}}^{{j\omega}_{2}{({t - \tau_{2}})}}} \right)^{*}\left( {{{\overset{\sim}{x}}_{1}\left( {t - \tau_{3}} \right)}{\mathbb{e}}^{{j\omega}_{1}{({t - \tau_{3}})}}} \right)} + {\left( {{{\overset{\sim}{x}}_{2}\left( {t - \tau_{1}} \right)}{\mathbb{e}}^{{j\omega}_{2}{({t - \tau_{1}})}}} \right)^{*}\left( {{{\overset{\sim}{x}}_{2}\left( {t - \tau_{2}} \right)}{\mathbb{e}}^{{j\omega}_{2}{({t - \tau_{2}})}}} \right)\left( {{{\overset{\sim}{x}}_{1}\left( {t - \tau_{3}} \right)}{\mathbb{e}}^{{j\omega}_{1}{({t - \tau_{3}})}}} \right)}}} & (13) \\{{\} \cdot {\mathbb{d}\tau_{3}}}{\mathbb{d}\tau_{2}}{\mathbb{d}\tau_{1}}} & (14)\end{matrix}$

Step 4: Continuous-time Passband to baseband equivalent transformation:In order to be implementable in a digital processor with manageablecomplexity, a pass band model should be transformed into a basebandequivalent model. Such a model enables mimicking the RF nonlineardynamic distortion while applying all the computations in baseband at alow sampling rate. The baseband equivalent model is obtained byfrequency translating the pass band Volterra series model to baseband.For that purpose, the continuous time pass band Volterra seriesexpressions of (13) and (14) are first rewritten in convolution form.The convolution form for) y_(ω) ₁ _(,1)(t) is given by:y _(ω) ₁ _(,1)(t)=h ₁(t)*({tilde over (x)} ₁(t)e ^(jω) ¹ ^(t))  (15)

As the kernel h₃ is tri-variate, h₃(τ₁, τ₂, τ₃), and the output y(t) ismono-variate, the output function is re-assigned as follows: y_(ω) ₁_(,3)(t)=y_(ω) ₁ _(,3)(t₁, t₂, t₃)_(|t) ₁ _(=t) ₂ _(=t) ₃ _(=t)

y_(ω) ₁ _(,3)(t, t, t). The convolution form is given below.

$\begin{matrix}{{y_{\omega_{1},3}\left( {t_{1},t_{2},t_{3}} \right)} = {{h_{3}\left( {t_{1},t_{2},t_{3}} \right)}*{\left\{ {{\left( {{{\overset{\sim}{x}}_{1}\left( t_{1} \right)}{\mathbb{e}}^{{j\omega}_{1}t_{1}}} \right)\left( {{{\overset{\sim}{x}}_{1}\left( t_{2} \right)}{\mathbb{e}}^{{j\omega}_{1}t_{2}}} \right)\left( {{{\overset{\sim}{x}}_{1}\left( t_{3} \right)}{\mathbb{e}}^{{j\omega}_{1}t_{3}}} \right)^{*}} + {\left( {{{\overset{\sim}{x}}_{1}\left( t_{1} \right)}{\mathbb{e}}^{{j\omega}_{1}t_{1}}} \right)\left( {{{\overset{\sim}{x}}_{1}\left( t_{2} \right)}{\mathbb{e}}^{{j\omega}_{1}t_{2}}} \right)\left( {{{\overset{\sim}{x}}_{1}\left( t_{3} \right)}{\mathbb{e}}^{{j\omega}_{1}t_{3}}} \right)} + {\left( {{{\overset{\sim}{x}}_{1}\left( t_{1} \right)}{\mathbb{e}}^{{j\omega}_{1}t_{1}}} \right)^{*}\left( {{{\overset{\sim}{x}}_{1}\left( t_{2} \right)}{\mathbb{e}}^{{j\omega}_{1}t_{2}}} \right)\left( {{{\overset{\sim}{x}}_{1}\left( t_{3} \right)}{\mathbb{e}}^{{j\omega}_{1}t_{3}}} \right)} + {\left( {{{\overset{\sim}{x}}_{1}\left( t_{1} \right)}{\mathbb{e}}^{{j\omega}_{1}t_{1}}} \right)\left( {{{\overset{\sim}{x}}_{2}\left( t_{2} \right)}{\mathbb{e}}^{{j\omega}_{2}t_{2}}} \right)\left( {{{\overset{\sim}{x}}_{2}\left( t_{3} \right)}{\mathbb{e}}^{{j\omega}_{2}t_{3}}} \right)^{*}} + {\left( {{{\overset{\sim}{x}}_{1}\left( t_{1} \right)}{\mathbb{e}}^{{j\omega}_{1}t_{1}}} \right)\left( {{{\overset{\sim}{x}}_{2}\left( t_{2} \right)}{\mathbb{e}}^{{j\omega}_{2}t_{2}}} \right)^{*}\left( {{{\overset{\sim}{x}}_{2}\left( t_{3} \right)}{\mathbb{e}}^{{j\omega}_{2}t_{3}}} \right)} + {\left( {{{\overset{\sim}{x}}_{2}\left( t_{1} \right)}{\mathbb{e}}^{{j\omega}_{2}t_{1}}} \right)^{*}\left( {{{\overset{\sim}{x}}_{1}\left( t_{2} \right)}{\mathbb{e}}^{{j\omega}_{1}t_{2}}} \right)\left( {{{\overset{\sim}{x}}_{2}\left( t_{3} \right)}{\mathbb{e}}^{{j\omega}_{2}t_{3}}} \right)} + {\left( {{\overset{\sim}{x}}_{2}\left( t_{1} \right){\mathbb{e}}^{{j\omega}_{2}t_{1}}} \right)^{*}\left( {{{\overset{\sim}{x}}_{1}\left( t_{2} \right)}{\mathbb{e}}^{{j\omega}_{1}t_{2}}} \right)\left( {{{\overset{\sim}{x}}_{2}\left( t_{3} \right)}{\mathbb{e}}^{{j\omega}_{2}t_{3}}} \right)} + {\left( {{{\overset{\sim}{x}}_{2}\left( t_{1} \right)}{\mathbb{e}}^{{j\omega}_{2}t_{1}}} \right)^{*}\left( {{{\overset{\sim}{x}}_{2}\left( t_{2} \right)}{\mathbb{e}}^{{j\omega}_{2}t_{2}}} \right)^{*}\left( {{{\overset{\sim}{x}}_{1}\left( t_{3} \right)}{\mathbb{e}}^{{j\omega}_{1}t_{3}}} \right)} + {\left( {{{\overset{\sim}{x}}_{2}\left( t_{1} \right)}{\mathbb{e}}^{{j\omega}_{2}t_{1}}} \right)^{*}\left( {{{\overset{\sim}{x}}_{2}\left( t_{2} \right)}{\mathbb{e}}^{{j\omega}_{2}t_{2}}} \right)\left( {{{\overset{\sim}{x}}_{1}\left( t_{3} \right)}{\mathbb{e}}^{{j\omega}_{1}t_{3}}} \right)}} \right\}.}}} & (16)\end{matrix}$

The application of the Laplace transform to (15) and (16) yield thefollowing expressions:

$\begin{matrix}\left. {Y_{\omega_{1},1} = {{\mathcal{L}\left( {y_{\omega_{1},1}(t)} \right)} = {{{\mathcal{L}\left( {h_{1}(t)} \right)}~ \cdot {\mathcal{L}\left( \left( {{{\overset{\sim}{x}}_{1}(t)}{\mathbb{e}}^{{j\omega}_{1}t}} \right) \right)}} = {{{H_{1}(s)}{{\overset{\sim}{X}}_{1}\left( {s - {j\omega}_{1}} \right)}{Y_{\omega_{1},3}\left( {s_{1},s_{2},s_{3}} \right)}} = {{\mathcal{L}\left( {y_{\omega_{1},3}\left( {t_{1},t_{2},t_{3}} \right)} \right)} = {{{\mathcal{L}\left( {h_{3}\left( {t_{1},t_{2},t_{3}} \right)} \right)} \cdot \left\{ {{\mathcal{L}\left( {\left( {{{\overset{\sim}{x}}_{1}\left( t_{1} \right)}{\mathbb{e}}^{{j\omega}_{1}t_{1}}} \right)\left( {{{\overset{\sim}{x}}_{1}\left( t_{2} \right)}{\mathbb{e}}^{{j\omega}_{1}t_{2}}} \right)\left( {{{\overset{\sim}{x}}_{1}\left( t_{2} \right)}{\mathbb{e}}^{{j\omega}_{1}t_{3}}} \right)^{*}} \right)} + {\mathcal{L}\left( {\left( {{{\overset{\sim}{x}}_{1}\left( t_{1} \right)}{\mathbb{e}}^{{j\omega}_{1}t_{1}}} \right)\left( {{{\overset{\sim}{x}}_{1}\left( t_{2} \right)}{\mathbb{e}}^{{j\omega}_{1}t_{2}}} \right)^{*}\left( {{{\overset{\sim}{x}}_{1}\left( t_{3} \right)}{\mathbb{e}}^{{j\omega}_{1}t_{3}}} \right)} \right)} + {\mathcal{L}\left( {\left( {{{\overset{\sim}{x}}_{1}\left( t_{1} \right)}{\mathbb{e}}^{{j\omega}_{1}t_{1}}} \right)^{*}\left( {{{\overset{\sim}{x}}_{1}\left( t_{2} \right)}{\mathbb{e}}^{{j\omega}_{1}t_{2}}} \right)\left( {{{\overset{\sim}{x}}_{1}\left( t_{3} \right)}{\mathbb{e}}^{{j\omega}_{1}t_{3}}} \right)} \right)} + {\mathcal{L}\left( {\left( {{{\overset{\sim}{x}}_{1}\left( t_{1} \right)}{\mathbb{e}}^{{j\omega}_{1}t_{1}}} \right)\left( {{{\overset{\sim}{x}}_{2}\left( t_{2} \right)}{\mathbb{e}}^{{j\omega}_{2}t_{2}}} \right)\left( {{{\overset{\sim}{x}}_{2}\left( t_{3} \right)}{\mathbb{e}}^{{j\omega}_{2}t_{3}}} \right)^{*}} \right)} + {\mathcal{L}\left( {\left( {{{\overset{\sim}{x}}_{1}\left( t_{1} \right)}{\mathbb{e}}^{{j\omega}_{1}t_{1}}} \right)\left( {{{\overset{\sim}{x}}_{2}\left( t_{2} \right)}{\mathbb{e}}^{{j\omega}_{2}t_{2}}} \right)^{*}\left( {{{\overset{\sim}{x}}_{2}\left( t_{3} \right)}{\mathbb{e}}^{{j\omega}_{2}t_{3}}} \right)} \right)} + {\mathcal{L}\left( {\left( {{{\overset{\sim}{x}}_{1}\left( t_{1} \right)}{\mathbb{e}}^{{j\omega}_{1}t_{1}}} \right)\left( {{{\overset{\sim}{x}}_{2}\left( t_{2} \right)}{\mathbb{e}}^{{j\omega}_{2}t_{2}}} \right)^{*}\left( {{{\overset{\sim}{x}}_{2}\left( t_{3} \right)}{\mathbb{e}}^{{j\omega}_{2}t_{3}}} \right)} \right)} + {\mathcal{L}\left( {\left( {{{\overset{\sim}{x}}_{2}\left( t_{1} \right)}{\mathbb{e}}^{{j\omega}_{1}t_{2}}} \right)^{*}\left( {{{\overset{\sim}{x}}_{1}\left( t_{2} \right)}{\mathbb{e}}^{{j\omega}_{1}t_{2}}} \right)\left( {{{\overset{\sim}{x}}_{2}\left( t_{3} \right)}{\mathbb{e}}^{{j\omega}_{2}t_{3}}} \right)} \right)} + {\mathcal{L}\left( {\left( {{{\overset{\sim}{x}}_{2}\left( t_{1} \right)}{\mathbb{e}}^{{j\omega}_{2}t_{1}}} \right)\left( {{{\overset{\sim}{x}}_{1}\left( t_{2} \right)}{\mathbb{e}}^{{j\omega}_{1}t_{2}}} \right)\left( {{{\overset{\sim}{x}}_{2}\left( t_{3} \right)}{\mathbb{e}}^{{j\omega}_{2}t_{3}}} \right)^{*}} \right)} + {\mathcal{L}\left( {\left( {{{\overset{\sim}{x}}_{2}\left( t_{1} \right)}{\mathbb{e}}^{{j\omega}_{2}t_{1}}} \right)\left( {{{\overset{\sim}{x}}_{2}\left( t_{2} \right)}{\mathbb{e}}^{{j\omega}_{2}t_{2}}} \right)^{*}\left( {{{\overset{\sim}{x}}_{1}\left( t_{3} \right)}{\mathbb{e}}^{{j\omega}_{1}t_{3}}} \right)} \right)} + {\mathcal{L}\left( {\left( {{{\overset{\sim}{x}}_{1}\left( t_{1} \right)}{\mathbb{e}}^{{j\omega}_{2}t_{1}}} \right)^{*}\left( {{{\overset{\sim}{x}}_{2}\left( t_{2} \right)}{\mathbb{e}}^{{j\omega}_{1}t_{3}}} \right)} \right)}} \right\}} = {{{\mathcal{L}\left( {h_{3}\left( {t_{1},t_{2},t_{3}} \right)} \right)} \cdot {\left\{ {{{\mathcal{L}\left( \left( {{{\overset{\sim}{x}}_{1}\left( t_{1} \right)}{\mathbb{e}}^{{j\omega}_{1}t_{1}}} \right) \right)}{\mathcal{L}\left( \left( {{{\overset{\sim}{x}}_{1}\left( t_{2} \right)}{\mathbb{e}}^{{j\omega}_{1}t_{2}}} \right) \right)}{\mathcal{L}\left( \left( {{{\overset{\sim}{x}}_{1}\left( t_{3} \right)}{\mathbb{e}}^{{j\omega}_{1}t_{3}}} \right)^{*} \right)}} + {{\mathcal{L}\left( \left( {{{\overset{\sim}{x}}_{1}\left( t_{1} \right)}{\mathbb{e}}^{{j\omega}_{1}t_{1}}} \right) \right)}{\mathcal{L}\left( \left( {{{\overset{\sim}{x}}_{1}\left( t_{1} \right)}{\mathbb{e}}^{{j\omega}_{1}t_{1}}} \right) \right)}{\mathcal{L}\left( \left( {{{\overset{\sim}{x}}_{1}\left( t_{2} \right)}{\mathbb{e}}^{{j\omega}_{1}t_{3}}} \right) \right)}} + {{\mathcal{L}\left( \left( {{{\overset{\sim}{x}}_{1}\left( t_{1} \right)}{\mathbb{e}}^{{j\omega}_{1}t_{1}}} \right)^{*} \right)}{\mathcal{L}\left( \left( {{{\overset{\sim}{x}}_{1}\left( t_{2} \right)}{\mathbb{e}}^{{j\omega}_{1}t_{2}}} \right) \right)}{\mathcal{L}\left( \left( {{{\overset{\sim}{x}}_{1}\left( t_{3} \right)}{\mathbb{e}}^{{j\omega}_{1}t_{3}}} \right) \right)}} + {{\mathcal{L}\left( \left( {{{\overset{\sim}{x}}_{1}\left( t_{1} \right)}{\mathbb{e}}^{{j\omega}_{1}t_{1}}} \right) \right)}{\mathcal{L}\left( \left( {{{\overset{\sim}{x}}_{2}\left( t_{2} \right)}{\mathbb{e}}^{{j\omega}_{2}t_{2}}} \right) \right)}{\mathcal{L}\left( \left( {{{\overset{\sim}{x}}_{2}\left( t_{3} \right)}{\mathbb{e}}^{{j\omega}_{2}t_{3}}} \right)^{*} \right)}} + {{\mathcal{L}\left( \left( {{{\overset{\sim}{x}}_{1}\left( t_{1} \right)}{\mathbb{e}}^{{j\omega}_{1}t_{1}}} \right) \right)}{\mathcal{L}\left( \left( {{{\overset{\sim}{x}}_{2}\left( t_{2} \right)}{\mathbb{e}}^{{j\omega}_{2}t_{2}}} \right)^{*} \right)}{\mathcal{L}\left( \left( {{{\overset{\sim}{x}}_{2}\left( t_{3} \right)}{\mathbb{e}}^{{j\omega}_{2}t_{3}}} \right) \right)}} + {{\mathcal{L}\left( \left( {{{\overset{\sim}{x}}_{2}\left( t_{1} \right)}{\mathbb{e}}^{{j\omega}_{2}t_{1}}} \right)^{*} \right)}{\mathcal{L}\left( \left( {{{\overset{\sim}{x}}_{1}\left( t_{2} \right)}{\mathbb{e}}^{{j\omega}_{1}t_{2}}} \right) \right)}{\mathcal{L}\left( \left( {{{\overset{\sim}{x}}_{2}\left( t_{3} \right)}{\mathbb{e}}^{{j\omega}_{2}t_{3}}} \right) \right)}} + {{\mathcal{L}\left( \left( {{{\overset{\sim}{x}}_{2}\left( t_{1} \right)}{\mathbb{e}}^{{j\omega}_{2}t_{1}}} \right) \right)}{\mathcal{L}\left( \left( {{{\overset{\sim}{x}}_{1}\left( t_{2} \right)}{\mathbb{e}}^{{j\omega}_{1}t_{2}}} \right) \right)}{\mathcal{L}\left( \left( {{{\overset{\sim}{x}}_{2}\left( t_{3} \right)}{\mathbb{e}}^{{j\omega}_{2}t_{3}}} \right)^{*} \right)}} + {{\mathcal{L}\left( \left( {{{\overset{\sim}{x}}_{2}\left( t_{1} \right)}{\mathbb{e}}^{{j\omega}_{2}t_{1}}} \right) \right)}{\mathcal{L}\left( \left( {{{\overset{\sim}{x}}_{2}\left( t_{2} \right)}{\mathbb{e}}^{{j\omega}_{2}t_{2}}} \right)^{*} \right)}{\mathcal{L}\left( \left( {{{\overset{\sim}{x}}_{1}\left( t_{3} \right)}{\mathbb{e}}^{{j\omega}_{1}t_{3}}} \right) \right)}} + {{\mathcal{L}\left( \left( {{{\overset{\sim}{x}}_{2}\left( t_{1} \right)}{\mathbb{e}}^{{j\omega}_{2}t_{1}}} \right)^{*} \right)}{\mathcal{L}\left( \left( {{{\overset{\sim}{x}}_{2}\left( t_{2} \right)}{\mathbb{e}}^{{j\omega}_{2}t_{2}}} \right) \right)}{\mathcal{L}\left( \left( {{{\overset{\sim}{x}}_{1}\left( t_{3} \right)}{\mathbb{e}}^{{j\omega}_{1}t_{3}}} \right) \right)}}} \right\}.}} = {{{H_{3}\left( {s_{1},s_{2},s_{3}} \right)} \cdot \left\{ {{{\overset{\sim}{X}}_{1}\left( {s_{1} - {j\omega}_{1}} \right)}{X_{1}^{*}\left( \left( {s_{3} - {j\omega}_{1}} \right)^{*} \right)}} \right)} + {{{\overset{\sim}{X}}_{1}\left( {s_{1} - {j\omega}_{1}} \right)}{{\overset{\sim}{X}}_{1}^{*}\left( \left( {s_{2} - {j\omega}_{1}} \right)^{*} \right)}{{\overset{\sim}{X}}_{1}\left( {s_{3} - {j\omega}_{1}} \right)}} + {{{\overset{\sim}{X}}_{1}^{*}\left( \left( {s_{1} - {j\omega}_{1}} \right)^{*} \right)}{{\overset{\sim}{X}}_{1}\left( {s_{2} - {j\omega}_{1}} \right)}{{\overset{\sim}{X}}_{1}\left( {s_{3} - {j\omega}_{1}} \right)}} + {{{\overset{\sim}{X}}_{1}\left( {s_{1} - {j\omega}_{1}} \right)}{{\overset{\sim}{X}}_{2}\left( {s_{2} - {j\omega}_{2}} \right)}{{\overset{\sim}{X}}_{2}^{*}\left( \left( {s_{3} - {j\omega}_{2}} \right)^{*} \right)}} + {{{\overset{\sim}{X}}_{1}\left( {s_{1} - {j\omega}_{1}} \right)}{{\overset{\sim}{X}}_{2}^{*}\left( \left( {s_{2} - {j\omega}_{2}} \right)^{*} \right)}{{\overset{\sim}{X}}_{2}\left( {s_{3} - {j\omega}_{2}} \right)}} + {{{\overset{\sim}{X}}_{2}^{*}\left( \left( {s_{1} - {j\omega}_{2}} \right)^{*} \right)}{{\overset{\sim}{X}}_{1}\left( {s_{2} - {j\omega}_{1}} \right)}{{\overset{\sim}{X}}_{2}\left( {s_{3} - {j\omega}_{2}} \right)}} + {{{\overset{\sim}{X}}_{2}\left( {s_{1} - {j\omega}_{2}} \right)}{{\overset{\sim}{X}}_{1}\left( {s_{2} - {j\omega}_{1}} \right)}{{\overset{\sim}{X}}_{2}^{*}\left( \left( {s_{3} - {j\omega}_{2}} \right)^{*} \right)}} + {{{\overset{\sim}{X}}_{2}\left( {s_{1} - {j\omega}_{2}} \right)}{{\overset{\sim}{X}}_{2}^{*}\left( \left( {s_{2} - {j\omega}_{2}} \right)^{*} \right)}{{\overset{\sim}{X}}_{1}\left( {s_{3} - {j\omega}_{1}} \right)}} + {{{\overset{\sim}{X}}_{2}^{*}\left( \left( {s_{1} - {j\omega}_{2}} \right)^{*} \right)}{{\overset{\sim}{X}}_{2}\left( {s_{2} - {j\omega}_{2}} \right)}{{\overset{\sim}{X}}_{1}\left( {s_{3} - {j\omega}_{1}} \right)}}}}}}}}}} \right\} & (17)\end{matrix}$

Since {tilde over (X)}₁ and {tilde over (X)}₂ are band limited signals,the third order distortion terms in (18) are non-zero only in a range ofa frequency intervals. For example, {tilde over (X)}₁(s₁−jω₁){tilde over(X)}₂(s₂−jω₂){tilde over (X)}₂*((s₃−jω₂)*) is non-zero only when s₁ εI₁, s₂ ε I₂, s₃ ε I₂, where

${I_{1} = \left\lbrack {{\omega_{1} - \frac{B}{2}},{\omega_{1} + \frac{B}{2}}} \right\rbrack};\;{I_{2 =}\left\lbrack {{\omega_{2} - \frac{B}{2}},{\omega_{2} + \frac{B}{2}}} \right\rbrack}$where B designates the bandwidth of the distortion term. Accordingly,one can redefine H₃(s₁, s₂, s₃) as follows:

${H_{3}\left( {s_{1},s_{2},s_{3}} \right)} = \left\{ \begin{matrix}{H_{3,\; s}\left( {s_{1},s_{2},s_{3}} \right)} & {{{{for}\mspace{14mu} s_{i}} \in I_{1}},{{i = 1};2;3}} \\{H_{3,{d\; 1}}\left( {s_{1},s_{2},s_{3}} \right)} & {{{{for}\mspace{14mu} s_{1}} \in I_{1}},{{{s_{i} \in \;{I_{2}i}} = 2};3}} \\{H_{3,{d\; 2}}\left( {{s_{1,}s_{2}},s_{3}} \right)} & {{{{for}\mspace{14mu} s_{2}} \in I_{1}},{{{s_{i} \in {I_{2}i}} = 1};3}} \\{H_{3,{d\; 3}}\left( {s_{1},s_{2},s_{3}} \right)} & {{{{for}\mspace{14mu} s_{3}} \in I_{1}},{{{s_{i} \in {I_{2}i}} = 1};2}}\end{matrix} \right.$Hence, (18) can be rewritten as

$\begin{matrix}{{{Y_{\omega_{1},3}\left( {s_{1},s_{2},s_{3}} \right)} = {{Y_{\omega_{1},3,1}\left( {s_{1},s_{2},s_{3}} \right)} + {Y_{\omega_{1},3,2}\left( {s_{1},s_{2},s_{3}} \right)} + {Y_{\omega_{1},3,3}\left( {s_{1},s_{2},s_{3}} \right)} + {Y_{\omega_{1},3,4}\left( {s_{1},s_{2},s_{3}} \right)}}}\mspace{79mu}{where}{{Y_{\omega_{1},3,1}\left( {s_{1},s_{2},s_{3}} \right)} = {{H_{3,s}\left( {s_{1},s_{2},s_{3}} \right)} \cdot \left\{ {{{{\overset{\sim}{X}}_{1}\left( {s_{1} - {j\omega}_{1}} \right)}{{\overset{\sim}{X}}_{1}\left( {s_{2} - {j\omega}_{1}} \right)}{{\overset{\sim}{X}}_{1}^{*}\left( \left( {s_{3} - {j\omega}_{1}} \right)^{*} \right)}} + {{{\overset{\sim}{X}}_{1}\left( {s_{1} - {j\omega}_{1}} \right)}{{\overset{\sim}{X}}_{1}^{*}\left( \left( {s_{2} - {j\omega}_{1}} \right)^{*} \right)}{{\overset{\sim}{X}}_{1}\left( {s_{3} - {j\omega}_{1}} \right)}} + {{{\overset{\sim}{X}}_{1}^{*}\left( \left( {s_{1} - {j\omega}_{1}} \right)^{*} \right)}{{\overset{\sim}{X}}_{1}\left( {s_{2} - {j\omega}_{1}} \right)}{{\overset{\sim}{X}}_{1}\left( {s_{3} - {j\omega}_{1}} \right)}}} \right\}}}{{Y_{\omega_{1},3,2}\left( {s_{1},s_{2},s_{3}} \right)} = {{H_{3,{d\; 1}}\left( {s_{1},s_{2},s_{3}} \right)} \cdot \left\{ {{{{\overset{\sim}{X}}_{1}\left( {s_{1} - {j\omega}_{1}} \right)}{{\overset{\sim}{X}}_{2}\left( {s_{2} - {j\omega}_{2}} \right)}{{\overset{\sim}{X}}_{2}^{*}\left( \left( {s_{3} - {j\omega}_{2}} \right)^{*} \right)}} + {{{\overset{\sim}{X}}_{1}\left( {s_{1} - {j\omega}_{1}} \right)}{{\overset{\sim}{X}}_{2}^{*}\left( \left( {s_{2} - {j\omega}_{2}} \right)^{*} \right)}{{\overset{\sim}{X}}_{2}\left( {s_{3} - {j\omega}_{2}} \right)}}} \right\}}}} & \; \\{{{Y_{\omega_{1},3,3}\left( {s_{1},s_{2},s_{3}} \right)} = {{H_{3,{d\; 2}}\left( {s_{1},s_{2},s_{3}} \right)} \cdot \left\{ {{{\overset{\sim}{X_{2}^{*}}\left( \left( {s_{1} - {j\omega}_{2}} \right)^{*} \right)}{{\overset{\sim}{X}}_{1}\left( {s_{2} - {j\omega}_{1}} \right)}{{\overset{\sim}{X}}_{2}\left( {s_{3} - {j\omega}_{2}} \right)}} + {{{\overset{\sim}{X}}_{2}\left( {s_{1} - {j\omega}_{2}} \right)}{{\overset{\sim}{X}}_{1}\left( {s_{2} - {j\omega}_{1}} \right)}{\overset{\sim}{X_{2}^{*}}\left( \left( {s_{3} - {j\omega}_{2}} \right)^{*} \right)}}} \right\}}}{{Y_{\omega_{1},3,4}\left( {s_{1},s_{2},s_{3}} \right)} = {{H_{3,{d\; 3}}\left( {s_{1},s_{2},s_{3}} \right)} \cdot \left\{ {{{\overset{\sim}{X_{2}}\left( {s_{1} - {j\omega}_{2}} \right)}{\overset{\sim}{X_{2}^{*}}\left( \left( {s_{2} - {j\omega}_{2}} \right)^{*} \right)}{{\overset{\sim}{X}}_{1}\left( {s_{3} - {j\omega}_{1}} \right)}} + {{\overset{\sim}{X_{2}^{*}}\left( \left( {s_{1} - {j\omega}_{2}} \right)^{*} \right)}{{\overset{\sim}{X}}_{2}\left( {s_{2} - {j\omega}_{2}} \right)}{\overset{\sim}{X_{1}}\left( {s_{3} - {j\omega}_{1}} \right)}}} \right\}}}} & (19)\end{matrix}$

Exploiting the symmetry of H_3 (s_1,s_2,s_3), yields the followingrelation H₃ (s₁, s₂, s₃)=H₃(s₂, s₁, s₃)=H₃(s₃, s₂, s₁) ∀s₁, s₂, s₃ fromwhich the following equalities can be deduced.H _(3,d1)(s ₁ ,s ₂ ,s ₃)=H _(3,d2)(s ₂ ,s ₁ ,s ₃)=H _(3,d3)(s ₃ ,s ₂ ,s₁)

Consequently a single kernel H_(3,d) can be used instead of the threeseparate ones where only the variable order is adjusted each time asshown below:H _(3,d1)(s ₁ ,s ₂ ,s ₃)=H _(3,d)(s ₁ ,s ₂ ,s ₃)H _(3,d2)(s ₁ ,s ₂ ,s ₃)=H _(3,d)(s ₂ ,s ₁ ,s ₃)  (20)H _(3,d3)(s ₁ ,s ₂ ,s ₃)=H _(3,d)(s ₃ ,s ₂ ,s ₁)Substituting (20) in (19) yields a new expression

$\begin{matrix}{{Y_{\omega_{1},3,1}\left( {s_{1},s_{2},s_{3}} \right)} = {{H_{3,s}\left( {s_{1},s_{2},s_{3}} \right)} \cdot \left\{ {{{{\overset{\sim}{X}}_{1}\left( {s_{1} - {j\omega}_{1}} \right)}{{\overset{\sim}{X}}_{1}\left( {s_{2} - {j\omega}_{1}} \right)}{{\overset{\sim}{X}}_{1}^{*}\left( \left( {s_{3} - {j\omega}_{1}} \right)^{*} \right)}} + {{{\overset{\sim}{X}}_{1}\left( {s_{1} - {j\omega}_{1}} \right)}{{\overset{\sim}{X}}_{1}^{*}\left( \left( {s_{2} - {j\omega}_{1}} \right)^{*} \right)}{{\overset{\sim}{X}}_{1}\left( {s_{3} - {j\omega}_{1}} \right)}} + {{{\overset{\sim}{X}}_{1}^{*}\left( \left( {s_{1} - {j\omega}_{1}} \right)^{*} \right)}{{\overset{\sim}{X}}_{1}\left( {s_{2} - {j\omega}_{1}} \right)}{{\overset{\sim}{X}}_{1}\left( {s_{3} - {j\omega}_{1}} \right)}}} \right\}}} & \; \\{{{Y_{\omega_{1},3,2}\left( {s_{1},s_{2},s_{3}} \right)} = {{H_{{3,d}\;}\left( {s_{1},s_{2},s_{3}} \right)} \cdot \left\{ {{{{\overset{\sim}{X}}_{1}\left( {s_{1} - {j\omega}_{1}} \right)}{{\overset{\sim}{X}}_{2}\left( {s_{2} - {j\omega}_{2}} \right)}{{\overset{\sim}{X}}_{2}^{*}\left( \left( {s_{3} - {j\omega}_{2}} \right)^{*} \right)}} + {{{\overset{\sim}{X}}_{1}\left( {s_{1} - {j\omega}_{1}} \right)}{{\overset{\sim}{X}}_{2}^{*}\left( \left( {s_{2} - {j\omega}_{2}} \right)^{*} \right)}{{\overset{\sim}{X}}_{2}\left( {s_{3} - {j\omega}_{2}} \right)}}} \right\}}}{{Y_{\omega_{1},3,3}\left( {s_{1},s_{2},s_{3}} \right)} = {{H_{{3,d}\;}\left( {s_{2},s_{1},s_{3}} \right)} \cdot \left\{ {{{\overset{\sim}{X_{2}^{*}}\left( \left( {s_{1} - {j\omega}_{2}} \right)^{*} \right)}{{\overset{\sim}{X}}_{1}\left( {s_{2} - {j\omega}_{1}} \right)}{{\overset{\sim}{X}}_{2}\left( {s_{3} - {j\omega}_{2}} \right)}} + {{{\overset{\sim}{X}}_{2}\left( {s_{1} - {j\omega}_{2}} \right)}{{\overset{\sim}{X}}_{1}\left( {s_{2} - {j\omega}_{1}} \right)}{\overset{\sim}{X_{2}^{*}}\left( \left( {s_{3} - {j\omega}_{2}} \right)^{*} \right)}}} \right\}}}{{Y_{\omega_{1},3,4}\left( {s_{1},s_{2},s_{3}} \right)} = {{H_{{3,d}\;}\left( {s_{3},s_{2},s_{1}} \right)} \cdot \left\{ {{{\overset{\sim}{X_{2}}\left( {s_{1} - {j\omega}_{2}} \right)}{\overset{\sim}{X_{2}^{*}}\left( \left( {s_{2} - {j\omega}_{2}} \right)^{*} \right)}{{\overset{\sim}{X}}_{1}\left( {s_{3} - {j\omega}_{1}} \right)}} + {{\overset{\sim}{X_{2}^{*}}\left( \left( {s_{1} - {j\omega}_{2}} \right)^{*} \right)}{{\overset{\sim}{X}}_{2}\left( {s_{2} - {j\omega}_{2}} \right)}{\overset{\sim}{X_{1}}\left( {s_{3} - {j\omega}_{1}} \right)}}} \right\}}}} & (21)\end{matrix}$

The application of a frequency translation in Laplace domain to (17) and(21) allows for passband to baseband equivalent transformation of y_(ω)₁ ^(,1) and y_(ω) ₁ _(,3)(t) in the time domain. The frequencytranslation of jω_(k) is performed by replacing s_(i) in (17) an (21)with u_(i)=s_(i)−jω_(k); i=1, 2, 3; k=1, 2. Hence, the application ofthe frequency translation to Y_(ω) ₁ _(,1)(s). Y_(ω) ₁ _(,3)(s₁, s₂,s₃). H₁(s), H_(3,s)(s₁, s₂, s₃), and H_(3,d)(s₁, s₂, s₃) yields thefollowing baseband equivalent expressions in the Laplace domain.

$\begin{matrix}{{{\overset{\sim}{Y}}_{\omega_{1},1}\left( u_{1} \right)} = {{Y_{\omega_{1},1}\left( {u_{1} + {j\omega}_{1}} \right)} = {{{H_{1}\left( {u_{1} + {j\omega}_{1}} \right)}{{\overset{\sim}{X}}_{1}\left( u_{1} \right)}} = {{{\overset{\sim}{H}}_{1}\left( u_{1} \right)}{\overset{\sim}{X}\left( u_{1} \right)}}}}} & (22) \\{{{\overset{\sim}{Y}}_{\omega_{1},3}\left( {u_{1},u_{2},u_{3}} \right)} = {{{\overset{\sim}{Y}}_{\omega_{1},3,1}\left( {u_{1},u_{2},u_{3}} \right)} + {{\overset{\sim}{Y}}_{\omega_{1},3,2}\left( {u_{1},u_{2},u_{3}} \right)} + {{\overset{\sim}{Y}}_{\omega_{1},3,3}\left( {u_{1},u_{2},u_{3}} \right)} + {{\overset{\sim}{Y}}_{\omega_{1},3,4}\left( {u_{1},u_{2},u_{3}} \right)}}} & (23) \\{\mspace{79mu}{{where}\text{:}}} & \; \\\begin{matrix}{\mspace{79mu}{{{\overset{\sim}{Y}}_{\omega_{1},3,1}\left( {u_{1},u_{2},u_{3}} \right)} = {Y_{\omega_{1},3,1}\left( {{u_{1} + {j\omega}_{1}},{u_{2} + {j\omega}_{1}},{u_{3} + {j\omega}_{1}}} \right)}}} \\{= {{H_{3,s}\left( {{u_{1} + {j\omega}_{1}},{u_{2} + {j\omega}_{1}},{u_{3} + {j\omega}_{1}}} \right)} \cdot}} \\{\left\{ {{{{\overset{\sim}{X}}_{1}\left( u_{1} \right)}{{\overset{\sim}{X}}_{1}\left( u_{2} \right)}{{\overset{\sim}{X}}_{1}^{*}\left( \left( u_{3} \right)^{*} \right)}} +} \right.} \\{{{{\overset{\sim}{X}}_{1}\left( u_{1} \right)}{{\overset{\sim}{X}}_{1}^{*}\left( \left( u_{2} \right)^{*} \right)}{{\overset{\sim}{X}}_{1}\left( u_{3} \right)}} +} \\\left. {{{\overset{\sim}{X}}_{1}^{*}\left( \left( u_{1} \right)^{*} \right)}{{\overset{\sim}{X}}_{1}\left( u_{2} \right)}{{\overset{\sim}{X}}_{1}\left( u_{3} \right)}} \right\} \\{= {{{\overset{\sim}{H}}_{3,s}\left( {u_{1},u_{2},u_{3}} \right)} \cdot}} \\{\left\{ {{{{\overset{\sim}{X}}_{1}\left( u_{1} \right)}{{\overset{\sim}{X}}_{1}\left( u_{2} \right)}{{\overset{\sim}{X}}_{1}^{*}\left( \left( u_{3} \right)^{*} \right)}} +} \right.} \\{{{{\overset{\sim}{X}}_{1}\left( u_{1} \right)}{{\overset{\sim}{X}}_{1}^{*}\left( \left( u_{2} \right)^{*} \right)}{{\overset{\sim}{X}}_{1}\left( u_{3} \right)}} +} \\\left. {{{\overset{\sim}{X}}_{1}^{*}\left( \left( u_{1} \right)^{*} \right)}{{\overset{\sim}{X}}_{1}\left( u_{2} \right)}{{\overset{\sim}{X}}_{1}\left( u_{3} \right)}} \right\}\end{matrix} & \; \\\begin{matrix}{\mspace{79mu}{{{\overset{\sim}{Y}}_{\omega_{1},3,2}\left( {u_{1},u_{2},u_{3}} \right)} = {Y_{\omega_{1},3,2}\left( {{u_{1} + {j\omega}_{1}},{u_{2} + {j\omega}_{2}},{u_{3} + {j\omega}_{2}}} \right)}}} \\{= {{H_{3,d}\left( {{u_{1} + {j\omega}_{1}},{u_{2} + {j\omega}_{2}},{u_{3} + {j\omega}_{2}}} \right)} \cdot}} \\{\left\{ {{{{\overset{\sim}{X}}_{1}\left( u_{1} \right)}{{\overset{\sim}{X}}_{2}\left( u_{2} \right)}{{\overset{\sim}{X}}_{2}^{*}\left( \left( u_{3} \right)^{*} \right)}} +} \right.} \\\left. {{{\overset{\sim}{X}}_{1}\left( u_{1} \right)}{{\overset{\sim}{X}}_{2}^{*}\left( \left( u_{2} \right)^{*} \right)}{{\overset{\sim}{X}}_{2}\left( u_{3} \right)}} \right\} \\{= {{{\overset{\sim}{H}}_{3,d}\left( {u_{1},u_{2},u_{3}} \right)} \cdot}} \\{\left\{ {{{{\overset{\sim}{X}}_{1}\left( u_{1} \right)}{{\overset{\sim}{X}}_{2}\left( u_{2} \right)}{{\overset{\sim}{X}}_{2}^{*}\left( \left( u_{3} \right)^{*} \right)}} +} \right.} \\\left. {{{\overset{\sim}{X}}_{1}\left( u_{1} \right)}{{\overset{\sim}{X}}_{2}^{*}\left( \left( u_{2} \right)^{*} \right)}{{\overset{\sim}{X}}_{2}\left( u_{3} \right)}} \right\}\end{matrix} & \; \\\begin{matrix}{\mspace{79mu}{{{\overset{\sim}{Y}}_{\omega_{1},3,3}\left( {u_{1},u_{2},u_{3}} \right)} = {Y_{\omega_{1},3,3}\left( {{u_{1} + {j\omega}_{2}},{u_{2} + {j\omega}_{1}},{u_{3} + {j\omega}_{2}}} \right)}}} \\{= {{H_{3,d}\left( {{u_{2} + {j\omega}_{1}},{u_{1} + {j\omega}_{2}},{u_{3} + {j\omega}_{2}}} \right)} \cdot}} \\{\left\{ {{{{\overset{\sim}{X}}_{2}^{*}\left( \left( u_{1} \right)^{*} \right)}{{\overset{\sim}{X}}_{1}\left( u_{2} \right)}{{\overset{\sim}{X}}_{2}\left( u_{3} \right)}} +} \right.} \\\left. \left. {{{\overset{\sim}{X}}_{2}\left( u_{1} \right)}{{\overset{\sim}{X}}_{1}\left( u_{2} \right)}{{\overset{\sim}{X}}_{2}^{*}\left( u_{3} \right)}^{*}} \right) \right\} \\{= {{{\overset{\sim}{H}}_{3,d}\left( {u_{2},u_{1},u_{3}} \right)} \cdot}} \\{\left\{ {{{{\overset{\sim}{X}}_{2}^{*}\left( \left( u_{1} \right)^{*} \right)}{{\overset{\sim}{X}}_{1}\left( u_{2} \right)}{{\overset{\sim}{X}}_{2}\left( u_{3} \right)}} +} \right.} \\\left. {{{\overset{\sim}{X}}_{2}\left( u_{1} \right)}{{\overset{\sim}{X}}_{1}\left( u_{2} \right)}{{\overset{\sim}{X}}_{2}^{*}\left( \left( u_{3} \right)^{*} \right)}} \right\}\end{matrix} & \; \\\begin{matrix}{\mspace{79mu}{{{\overset{\sim}{Y}}_{\omega_{1},3,4}\left( {u_{1},u_{2},u_{3}} \right)} = {Y_{\omega_{1},3,4}\left( {{u_{1} + {j\omega}_{2}},{u_{2} + {j\omega}_{2}},{u_{3} + {j\omega}_{1}}} \right)}}} \\{= {{H_{3,d}\left( {{u_{3} + {j\omega}_{1}},{u_{2} + {j\omega}_{2}},{u_{1} + {j\omega}_{2}}} \right)} \cdot}} \\{\left\{ {{{{\overset{\sim}{X}}_{2}\left( u_{1} \right)}{{\overset{\sim}{X}}_{2}^{*}\left( \left( u_{2} \right)^{*} \right)}{{\overset{\sim}{X}}_{1}\left( u_{3} \right)}} +} \right.} \\\left. {{{\overset{\sim}{X}}_{2}^{*}\left( \left( u_{1} \right)^{*} \right)}{{\overset{\sim}{X}}_{2}\left( u_{2} \right)}{{\overset{\sim}{X}}_{1}\left( u_{3} \right)}} \right\} \\{= {{{\overset{\sim}{H}}_{3,d}\left( {u_{3},u_{2},u_{1}} \right)} \cdot}} \\{\left\{ {{{{\overset{\sim}{X}}_{2}\left( u_{1} \right)}{{\overset{\sim}{X}}_{2}^{*}\left( \left( u_{2} \right)^{*} \right)}{{\overset{\sim}{X}}_{1}\left( u_{3} \right)}} +} \right.} \\\left. {{{\overset{\sim}{X}}_{2}^{*}\left( \left( u_{1} \right)^{*} \right)}{{\overset{\sim}{X}}_{2}\left( u_{2} \right)}{{\overset{\sim}{X}}_{1}\left( u_{3} \right)}} \right\}\end{matrix} & \;\end{matrix}$

The application of the inverse Laplace to (22) and (23) yields thefollowing time domain expressions of the baseband equivalent terms:

$\begin{matrix}{{\left. {{{{\overset{\sim}{y}}_{\omega_{1},1}(t)} = {\int_{- \infty}^{\infty}{{{\overset{\sim}{h}}_{1}\ \left( \tau_{1} \right)} \cdot {{\overset{\sim}{x}}_{1}\left( {t - \tau_{1}} \right)} \cdot {\mathbb{d}\tau_{1}}}}}{{{\overset{\sim}{y}}_{\omega_{1},3,1}(t)} = {\int_{- \infty}^{\infty}{\int_{- \infty}^{\infty}{\int_{- \infty}^{\infty}{{{{\overset{\sim}{h}}_{3,s}\left( {\tau_{1},\tau_{2},\tau_{3}} \right)} \cdot \left\{ {{\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{1}} \right)} \right)\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{2}} \right)} \right)\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{3}} \right)} \right)^{*}} + {\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{1}} \right)} \right)\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{2}} \right)} \right)^{*}\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{3}} \right)} \right)} + {\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{1}} \right)} \right)^{*}\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{2}} \right)} \right)\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{3}} \right)} \right)}} \right\} \cdot {\mathbb{d}\tau_{3}}}{\mathbb{d}\tau_{2}}{\mathbb{d}\tau_{1}}}}}}}{{{\overset{\sim}{y}}_{\omega_{1},3,2}(t)} = {\int_{- \infty}^{\infty}{\int_{- \infty}^{\infty}{\int_{- \infty}^{\infty}{{\overset{\sim}{h}}_{3,d}\ {\left( {\tau_{1},\tau_{2},\tau_{3}} \right) \cdot \left\{ {{\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{1}} \right)} \right)\left( {{\overset{\sim}{x}}_{2}\left( {t - \tau_{2}} \right)} \right)^{*}} + \left( {{\overset{\sim}{x}}_{2}\left( {t - \tau_{3}} \right)} \right) + {\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{1}} \right)} \right)\left( {{\overset{\sim}{x}}_{2}\left( {t - \tau_{2}} \right)} \right)^{*}\left( {{\overset{\sim}{x}}_{2}\left( {t - \tau_{3}} \right)} \right)}} \right\} \cdot {\mathbb{d}\tau_{3}}}{\mathbb{d}\tau_{2}}{\mathbb{d}\tau_{1}}}}}}}{{{\overset{\sim}{y}}_{\omega_{1},3,3}(t)} = {{\int_{- \infty}^{\infty}{\int_{- \infty}^{\infty}{\int_{- \infty}^{\infty}{{\overset{\sim}{h}}_{3,d}\ {\left( {\tau_{2},\tau_{1},\tau_{3}} \right) \cdot \left\{ {{{+ \left( {{\overset{\sim}{x}}_{2}\left( {t - \tau_{1}} \right)} \right)^{*}}\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{2}} \right)} \right)\left( {{\overset{\sim}{x}}_{2}\left( {t - \tau_{3}} \right)} \right)} + {\left( {{\overset{\sim}{x}}_{2}\left( {t - \tau_{1}} \right)} \right)\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{2}} \right)} \right)\left( {{\overset{\sim}{x}}_{2}\left( {t - \tau_{3}} \right)} \right)^{*}}} \right\} \cdot {\mathbb{d}\tau_{3}}}{\mathbb{d}\tau_{2}}{\mathbb{d}\tau_{1}}{{\overset{\sim}{y}}_{\omega_{1},3,4}(t)}}}}} = {\int_{- \infty}^{\infty}{\int_{- \infty}^{\infty}{\int_{- \infty}^{\infty}{{\overset{\sim}{h}}_{3,d}\ {\left( {\tau_{3},\tau_{2},\tau_{1}} \right) \cdot \left\{ {{{+ \left( {{\overset{\sim}{x}}_{2}\left( {t - \tau_{1}} \right)} \right)}\left( {{\overset{\sim}{x}}_{2}\left( {t - \tau_{2}} \right)} \right)^{*}\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{3}} \right)} \right)} + {\left( {{\overset{\sim}{x}}_{2}\left( {t - \tau_{1}} \right)} \right)^{*}\left( {{\overset{\sim}{x}}_{2}\left( {t - \tau_{2}} \right)} \right)\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{3}} \right)} \right)}} \right\}}}}}}}}} \right\} \cdot {\mathbb{d}\tau_{3}}}{\mathbb{d}\tau_{2}}{\mathbb{d}\tau_{1}}} & (24)\end{matrix}$

Swapping τ₂with τ₁ in {tilde over (y)}_(ω) ₁ _(,3,3)(t) and τ₃with τ₁ in{tilde over (y)}_(ω) ₁ _(,3,4) yields the following equality{tilde over (y)} _(ω) ₁ _(,3,2)(t)={tilde over (y)} _(ω) ₁_(,3,3)(t)={tilde over (y)} _(ω) ₁ _(,3,4)(t)Hence, the third order baseband equivalent Volterra term could bere-written as:

$\begin{matrix}\begin{matrix}{{{\overset{\sim}{y}}_{\omega_{1},3}(t)} = {{{\overset{\sim}{y}}_{\omega_{1},3,1}(t)} + {3\;{{\overset{\sim}{y}}_{\omega_{1},3,2}(t)}}}} \\{= {{{\overset{\sim}{y}}_{\omega_{1},3,s}(t)} + {{\overset{\sim}{y}}_{\omega_{1},3,d}(t)}}}\end{matrix} & (25)\end{matrix}$where {tilde over (y)}_(ω) ₁ _(,3,s)(t)={tilde over (y)}_(ω) ₁_(,3,1)(t) designates the PA third order single-band self-distortionterm 20 and {tilde over (y)}_(ω) ₁ _(,3,d)(t)=3{tilde over (y)}_(ω) ₁_(,3,2)(t) denotes the PA third order dual-band inter-band-distortionterm 22, as shown in FIG. 5. These terms are, collectively, the basebandequivalent Volterra series which, when discretized as explained below,model the pre-distortion of a dual band power amplifier.

The same derivations were applied to construct the fifth order Volterradistortion term expression which is found to be

$\begin{matrix}{{{{\overset{\sim}{y}}_{\omega_{1},5}(t)} = {{{\overset{\sim}{y}}_{\omega_{1},5,s}(t)} + {{\overset{\sim}{y}}_{\omega_{1},5,{d1}}(t)} + {{\overset{\sim}{y}}_{\omega_{1},{5{d2}}}(t)}}}{where}\text{}{{{\overset{\sim}{y}}_{\omega_{1},5,s}(t)} = {\int_{- \infty}^{\infty}{\int_{- \infty}^{\infty}{\int_{- \infty}^{\infty}{\int_{- \infty}^{\infty}{\int_{- \infty}^{\infty}{{{{\overset{\sim}{h}}_{5,s}\left( {\tau_{1},\tau_{2},\tau_{3},\tau_{4},\tau_{5}} \right)} \cdot \left\{ {{\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{1}} \right)} \right)\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{2}} \right)} \right)\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{3}} \right)} \right)\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{4}} \right)} \right)^{*}\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{5}} \right)} \right)^{*}} + {\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{1}} \right)} \right)\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{2}} \right)} \right)\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{3}} \right)} \right)^{*}\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{4}} \right)} \right)\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{5}} \right)} \right)^{*}} + {\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{1}} \right)} \right)\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{2}} \right)} \right)^{*}\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{3}} \right)} \right)\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{4}} \right)} \right)\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{5}} \right)} \right)^{*}} + {\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{1}} \right)} \right)^{*}\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{2}} \right)} \right)\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{3}} \right)} \right)\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{4}} \right)} \right)\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{5}} \right)} \right)^{*}} + {\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{1}} \right)} \right)\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{2}} \right)} \right)\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{3}} \right)} \right)^{*}\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{4}} \right)} \right)^{*}\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{1}} \right)} \right)} + {\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{1}} \right)} \right)\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{2}} \right)} \right)^{*}\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{3}} \right)} \right)\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{4}} \right)} \right)^{*}\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{5}} \right)} \right)} + {\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{1}} \right)} \right)^{*}\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{2}} \right)} \right)\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{3}} \right)} \right)\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{4}} \right)} \right)^{*}\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{5}} \right)} \right)} + {\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{1}} \right)} \right)\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{2}} \right)} \right)^{*}\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{3}} \right)} \right)^{*}\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{4}} \right)} \right)\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{5}} \right)} \right)} + {\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{1}} \right)} \right)^{*}\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{2}} \right)} \right)\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{3}} \right)} \right)^{*}\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{4}} \right)} \right)\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{5}} \right)} \right)} + {\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{1}} \right)} \right)^{*}\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{2}} \right)} \right)^{*}\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{3}} \right)} \right)\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{4}} \right)} \right)\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{5}} \right)} \right)}} \right\} \cdot {\mathbb{d}\tau_{5}}}{\mathbb{d}\tau_{4}}{\mathbb{d}\tau_{3}}{\mathbb{d}\tau_{2}}{\mathbb{d}\tau_{1}}}}}}}}}{{{\overset{\sim}{y}}_{\omega_{1},5,{d\; 1}}(t)} = {\int_{- \infty}^{\infty}{\int_{- \infty}^{\infty}{\int_{- \infty}^{\infty}{\int_{- \infty}^{\infty}{\int_{- \infty}^{\infty}{{{{\overset{\sim}{h}}_{5,{d\; 1}}\left( {\tau_{1},\tau_{2},\tau_{3},\tau_{4},\tau_{5}} \right)} \cdot \left\{ {{{+ \left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{1}} \right)} \right)}\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{2}} \right)} \right)^{*}\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{3}} \right)} \right)\left( {{\overset{\sim}{x}}_{2}\left( {t - \tau_{4}} \right)} \right)\left( {{\overset{\sim}{x}}_{2}\left( {t - \tau_{5}} \right)} \right)^{*}} + {\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{1}} \right)} \right)^{*}\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{2}} \right)} \right)\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{3}} \right)} \right)\left( {{\overset{\sim}{x}}_{2}\left( {t - \tau_{4}} \right)} \right)\left( {{\overset{\sim}{x}}_{2}\left( {t - \tau_{5}} \right)} \right)^{*}} + {\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{1}} \right)} \right)\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{2}} \right)} \right)\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{3}} \right)} \right)^{*}\left( {{\overset{\sim}{x}}_{2}\left( {t - \tau_{4}} \right)} \right)^{*}\left( {{\overset{\sim}{x}}_{2}\left( {t - \tau_{5}} \right)} \right)} + {\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{1}} \right)} \right)\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{2}} \right)} \right)^{*}\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{3}} \right)} \right)\left( {{\overset{\sim}{x}}_{2}\left( {t - \tau_{4}} \right)} \right)^{*}\left( {{\overset{\sim}{x}}_{2}\left( {t - \tau_{5}} \right)} \right)} + {\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{1}} \right)} \right)^{*}\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{2}} \right)} \right)\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{3}} \right)} \right)\left( {{\overset{\sim}{x}}_{2}\left( {t - \tau_{4}} \right)} \right)^{*}\left( {{\overset{\sim}{x}}_{2}\left( {t - \tau_{5}} \right)} \right)}} \right\} \cdot {\mathbb{d}\tau_{5}}}{\mathbb{d}\tau_{4}}{\mathbb{d}\tau_{3}}{\mathbb{d}\tau_{2}}{\mathbb{d}\tau_{1}}}}}}}}}{{{\overset{\sim}{y}}_{\omega_{1},5,{d\; 2}}(t)} = {\int_{- \infty}^{\infty}{\int_{- \infty}^{\infty}{\int_{- \infty}^{\infty}{\int_{- \infty}^{\infty}{\int_{- \infty}^{\infty}{{{{\overset{\sim}{h}}_{5,{d\; 1}}\left( {\tau_{1},\tau_{2},\tau_{3},\tau_{4},\tau_{5}} \right)} \cdot \left\{ {{\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{1}} \right)} \right)\left( {{\overset{\sim}{x}}_{2}\left( {t - \tau_{2}} \right)} \right)\left( {{\overset{\sim}{x}}_{2}\left( {t - \tau_{3}} \right)} \right)\left( {{\overset{\sim}{x}}_{2}\left( {t - \tau_{4}} \right)} \right)^{*}\left( {{\overset{\sim}{x}}_{2}\left( {t - \tau_{5}} \right)} \right)^{*}} + {\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{1}} \right)} \right)\left( {{\overset{\sim}{x}}_{2}\left( {t - \tau_{2}} \right)} \right)\left( {{\overset{\sim}{x}}_{2}\left( {t - \tau_{3}} \right)} \right)^{*}\left( {{\overset{\sim}{x}}_{2}\left( {t - \tau_{3}} \right)} \right)\left( {{\overset{\sim}{x}}_{2}\left( {t - \tau_{4}} \right)} \right)\left( {{\overset{\sim}{x}}_{2}\left( {t - \tau_{5}} \right)} \right)^{*}} + {\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{1}} \right)} \right)\left( {{\overset{\sim}{x}}_{2}\left( {t - \tau_{2}} \right)} \right)^{*}\left( {{\overset{\sim}{x}}_{2}\left( {t - \tau_{3}} \right)} \right)\left( {{\overset{\sim}{x}}_{2}\left( {t - \tau_{4}} \right)} \right)\left( {{\overset{\sim}{x}}_{2}\left( {t - \tau_{5}} \right)} \right)^{*}} + {\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{1}} \right)} \right)\left( {{\overset{\sim}{x}}_{2}\left( {t - \tau_{2}} \right)} \right)\left( {{\overset{\sim}{x}}_{2}\left( {t - \tau_{3}} \right)} \right)^{*}\left( {{\overset{\sim}{x}}_{2}\left( {t - \tau_{4}} \right)} \right)^{*}\left( {{\overset{\sim}{x}}_{2}\left( {t - \tau_{5}} \right)} \right)} + {\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{1}} \right)} \right)\left( {{\overset{\sim}{x}}_{2}\left( {t - \tau_{2}} \right)} \right)^{*}\left( {{\overset{\sim}{x}}_{2}\left( {t - \tau_{3}} \right)} \right)\left( {{\overset{\sim}{x}}_{2}\left( {t - \tau_{4}} \right)} \right)^{*}\left( {{\overset{\sim}{x}}_{2}\left( {t - \tau_{5}} \right)} \right)} + {\left( {{\overset{\sim}{x}}_{1}\left( {t - \tau_{1}} \right)} \right)\left( {{\overset{\sim}{x}}_{2}\left( {t - \tau_{2}} \right)} \right)^{*}\left( {{\overset{\sim}{x}}_{2}\left( {t - \tau_{3}} \right)} \right)^{*}\left( {{\overset{\sim}{x}}_{2}\left( {t - \tau_{4}} \right)} \right)\left( {{\overset{\sim}{x}}_{2}\left( {t - \tau_{5}} \right)} \right)}} \right\} \cdot {\mathbb{d}\tau_{5}}}{\mathbb{d}\tau_{4}}{\mathbb{d}\tau_{3}}{\mathbb{d}\tau_{2}}{\mathbb{d}\tau_{1}}}}}}}}}} & (26)\end{matrix}$Hence, the continuous-time dual-band baseband equivalent Volterra seriesmodel for each band is given by:{tilde over (y)} _(ω) ₁ (t)={tilde over (y)} _(ω) ₁ _(,1)(t)+{tilde over(y)} _(ω) ₁ _(,3)(t)+{tilde over (y)} _(ω) ₁ _(,5)(t)+   (27)

Step 5: Discrete-time baseband equivalent Volterra series model: Inorder to implement the dual band BBE Volterra model in a digitalprocessor, the following signal and systems properties andapproximations are used to further simplify (27).

-   1. Truncation of the Volterra model to a finite nonlinearity order    NL, generally in the range of 5 to 7.-   2. Limitation of the integral bounds (−∞, +∞) to (0, T_(∞)) using    the signal and system causality, and the fading memory assumption    (transient response time invariant Volterra series is defined as    t<T_(∞)) [43]. Since the impulse responses of different Volterra    kernels, i.e. {tilde over (h)}_(ω) ₁ _(,1){tilde over (h)}_(ω) ₁    _(,3,s),{tilde over (h)}_(ω) ₁ _(,3,d), {tilde over (h)}_(ω) ₁    _(,5,s), . . . represent different aspects of the system, the memory    spans used in the computation of the different distortion terms can    be set to be different.-   3. Using the symmetry of the terms inside the integral (Distortion    components are symmetrical and Volterra kernels can be symmetrized),    the number of required kernels is significantly reduced.

Digitizing the dual band BBE Volterra model yields:

$\begin{matrix}{{{\overset{\sim}{y}}_{\omega_{1},1}(n)} = {{{\overset{\sim}{y}}_{\omega_{1},1}(n)} + {{\overset{\sim}{y}}_{\omega_{1},3}(n)} + {{\overset{\sim}{y}}_{\omega_{1},5}(n)} + \cdots}} & (28) \\{{{\overset{\sim}{y}}_{\omega_{1},1}(n)} = {\sum\limits_{i_{1} = 0}^{M_{1}}\;{{{\overset{\sim}{h}}_{\omega_{1},1}\left( i_{1} \right)}{{\overset{\sim}{x}}_{1,s}\left( {n,i_{1}} \right)}}}} & \; \\{{{\overset{\sim}{y}}_{\omega_{1},3}(n)} = {{\sum\limits_{i_{1} = 0}^{M_{3,s}}{\sum\limits_{i_{2} = i_{1}}^{M_{3,s}}{\sum\limits_{i_{3} = i_{2}}^{M_{3,s}}\;{{{\overset{\sim}{h}}_{\omega_{1},3,s}\left( {{i_{1,}i_{2}},i_{3}} \right)} \cdot {{\overset{\sim}{x}}_{3,s}\left( {n,{i_{1,}i_{2}},i_{3}} \right)}}}}} + {\sum\limits_{i_{1} = 0}^{M_{3,d}}{\sum\limits_{i_{2} = i_{1}}^{M_{3,d}}{\sum\limits_{i_{3} = i_{2}}^{M_{3,d}}{{{\overset{\sim}{h}}_{\omega_{1},3,d}\left( {{i_{1,}i_{2}},i_{3}} \right)} \cdot {{\overset{\sim}{x}}_{3,d}\left( {n,{i_{1,}i_{2}},i_{3}} \right)}}}}}}} & \; \\{{{\overset{\sim}{y}}_{\omega_{1},5}(n)} = {{\sum\limits_{i_{1} = 0}^{M_{5,s}}{\sum\limits_{i_{2} = i_{1}}^{M_{5,s}}{\sum\limits_{i_{3} = i_{2}}^{M_{5,s}}{\sum\limits_{i_{4} = i_{3}}^{M_{5,s}}{\sum\limits_{i_{5} = i_{4}}^{M_{5,s}}{{{\overset{\sim}{h}}_{\omega_{1},5,s}\left( {{i_{1,}i_{2}},i_{3},i_{4},i_{5}} \right)} \cdot {{\overset{\sim}{x}}_{5,s}\left( {n,{i_{1,}i_{2}},i_{3},i_{4},i_{5}} \right)}}}}}}} + {\sum\limits_{i_{1} = 0}^{M_{5,{d\; 1}}}{\sum\limits_{i_{2} = i_{1}}^{M_{5,{d\; 1}}}{\sum\limits_{i_{3} = i_{2}}^{M_{5,{d\; 1}}}{\sum\limits_{i_{4} = i_{3}}^{M_{5,{d\; 1}}}{\sum\limits_{i_{5} = i_{4}}^{M_{5,{d\; 1}}}{{{\overset{\sim}{h}}_{\omega_{1},5,{d\; 1}}\left( {{i_{1,}i_{2}},i_{3},i_{4},i_{5}} \right)} \cdot {{\overset{\sim}{x}}_{5,{d\; 1}}\left( {n,i_{1},i_{2},i_{3},i_{4},i_{5}} \right)}}}}}}} + {\sum\limits_{i_{1} = 0}^{M_{5,{d\; 2}}}{\sum\limits_{i_{2} = 0}^{M_{5,{d\; 2}}}{\sum\limits_{i_{3} = i_{2}}^{M_{5,{d\; 2}}}{\sum\limits_{i_{4} = i_{3}}^{M_{5,{d\; 2}}}{\sum\limits_{i_{5} = i_{4}}^{M_{5,{d\; 2}}}{{{\overset{\sim}{h}}_{\omega_{1},5,{d\; 2}}\left( {{i_{1,}i_{2}},i_{3},i_{4},i_{5}} \right)} \cdot {{\overset{\sim}{x}}_{5,{d\; 2}}\left( {n,i_{1},i_{2},i_{3},i_{4},i_{5}} \right)}}}}}}}}} & \; \\{where} & \; \\{{{\overset{\sim}{x}}_{1,s}\left( {n,i_{1}} \right)} = {\overset{\sim}{x}\left( {n - i_{1}} \right)}} & \; \\{{{\overset{\sim}{x}}_{3,s}\left( {n,i_{1},i_{2},i_{3}} \right)} = {{{\overset{\sim}{x_{1}}\left( {n - i_{1}} \right)}{{\overset{\sim}{x}}_{1}\left( {n - i_{2}} \right)}{{\overset{\sim}{x}}_{1}^{*}\left( {n - i_{3}} \right)}} + {{{\overset{\sim}{x}}_{1}^{*}\left( {n - i_{1}} \right)}{{\overset{\sim}{x}}_{1}\left( {n - i_{2}} \right)}{{\overset{\sim}{x}}_{1}\left( {n - i_{3}} \right)}}}} & \; \\{{{\overset{\sim}{x}}_{3,d}\left( {n,i_{1},i_{2},i_{2}} \right)} = {{{{\overset{\sim}{x}}_{1}\left( {n - i_{1}} \right)}{{\overset{\sim}{x}}_{2}\left( {n - i_{2}} \right)}{{\overset{\sim}{x}}_{2}^{*}\left( {n - i_{2}} \right)}} + {{{\overset{\sim}{x}}_{1}\left( {n - i_{i}} \right)}{{\overset{\sim}{x}}_{2}^{*}\left( {n - i_{2}} \right)}{{\overset{\sim}{x}}_{2}\left( {n - i_{2}} \right)}}}} & \; \\{{{\overset{\sim}{x}}_{5,s}\left( {n,i_{1},i_{2},i_{3},i_{4},i_{5}} \right)} = {{{{\overset{\sim}{x}}_{1}\left( {n - i_{1}} \right)}{{\overset{\sim}{x}}_{1}\left( {n - i_{2}} \right)}{{\overset{\sim}{x}}_{1}\left( {n - i_{3}} \right)}{{\overset{\sim}{x}}_{1}^{*}\left( {n - i_{4}} \right)}{{\overset{\sim}{x}}_{1}^{*}\left( {n - i_{s}} \right)}} + {{{\overset{\sim}{x}}_{1}\left( {n - i_{1}} \right)}{{\overset{\sim}{x}}_{1}\left( {n - i_{2}} \right)}{{\overset{\sim}{x}}_{1}^{*}\left( {n - i_{2}} \right)}{{\overset{\sim}{x}}_{1}^{*}\left( {n - i_{3}} \right)}{{\overset{\sim}{x}}_{1}\left( {n - i_{4}} \right)}{{\overset{\sim}{x}}_{1}^{*}\left( {n - i_{5}} \right)}} + {{{\overset{\sim}{x}}_{1}\left( {n - i_{1}} \right)}{{\overset{\sim}{x}}_{1}^{*}\left( {n - i_{2}} \right)}{{\overset{\sim}{x}}_{1}\left( {n - i_{3}} \right)}{{\overset{\sim}{x}}_{1}\left( {n - i_{4}} \right)}{{\overset{\sim}{x}}_{1}^{*}\left( {n - i_{5}} \right)}} + {{{\overset{\sim}{x}}_{1}\left( {n - i_{1}} \right)}{{\overset{\sim}{x}}_{1}\left( {n - i_{2}} \right)}{{\overset{\sim}{x}}_{1}^{*}\left( {n - i_{3}} \right)}{{\overset{\sim}{x}}_{1}^{*}\left( {n - i_{4}} \right)}{{\overset{\sim}{x}}_{1}\left( {n - i_{5}} \right)}} + {{{\overset{\sim}{x}}_{1}^{*}\left( {n - i_{1}} \right)}{{\overset{\sim}{x}}_{1}^{*}\left( {n - i_{2}} \right)}{{\overset{\sim}{x}}_{1}^{*}\left( {n - i_{3}} \right)}{{\overset{\sim}{x}}_{1}^{*}\left( {n - i_{4}} \right)}{{\overset{\sim}{x}}_{1}\left( {n - i_{5}} \right)}} + {{{\overset{\sim}{x}}_{1}\left( {n - i_{1}} \right)}{{\overset{\sim}{x}}_{1}^{*}\left( {n - i_{2}} \right)}{{\overset{\sim}{x}}_{1}^{*}\left( {n - i_{3}} \right)}{{\overset{\sim}{x}}_{1}\left( {n - i_{4}} \right)}{{\overset{\sim}{x}}_{1}\left( {n - i_{5}} \right)}} + {{{\overset{\sim}{x}}_{1}^{*}\left( {n - i_{1}} \right)}{{\overset{\sim}{x}}_{1}\left( {n - i_{2}} \right)}{{\overset{\sim}{x}}_{1}^{*}\left( {n - i_{3}} \right)}{{\overset{\sim}{x}}_{1}\left( {n - i_{4}} \right)}{{\overset{\sim}{x}}_{1}\left( {n - i_{5}} \right)}} + {{{\overset{\sim}{x}}_{1}^{*}\left( {n - i_{1}} \right)}{{\overset{\sim}{x}}_{1}^{*}\left( {n - i_{2}} \right)}{{\overset{\sim}{x}}_{1}\left( {n - i_{3}} \right)}{{\overset{\sim}{x}}_{1}\left( {n - i_{4}} \right)}{{\overset{\sim}{x}}_{1}\left( {n - i_{5}} \right)}}}} & \; \\{{{\overset{\sim}{x}}_{5,{d\; 1}}\left( {n,i_{1},i_{2},i_{3},i_{4},i_{5}} \right)} = {{{{\overset{\sim}{x}}_{1}\left( {n - i_{1}} \right)}{{\overset{\sim}{x}}_{1}\left( {n - i_{2}} \right)}{{\overset{\sim}{x}}_{1}^{*}\left( {n - i_{3}} \right)}{{\overset{\sim}{x}}_{2}\left( {n - i_{4}} \right)}{{\overset{\sim}{x}}_{1}\left( {n - i_{5}} \right)}} + {{{\overset{\sim}{x}}_{1}\left( {n - i_{1}} \right)}{{\overset{\sim}{x}}_{1}^{*}\left( {n - i_{2}} \right)}{{\overset{\sim}{x}}_{2}\left( {n - i_{3}} \right)}{{\overset{\sim}{x}}_{2}\left( {n - i_{4}} \right)}{{\overset{\sim}{x}}_{2}^{*}\left( {n - i_{5}} \right)}} + {{{\overset{\sim}{x}}_{1}^{*}\left( {n - i_{1}} \right)}{{\overset{\sim}{x}}_{1}\left( {n - i_{2}} \right)}{{\overset{\sim}{x}}_{1}\left( {n - i_{3}} \right)}{{\overset{\sim}{x}}_{2}\left( {n - i_{4}} \right)}{{\overset{\sim}{x}}_{2}^{*}\left( {n - i_{5}} \right)}} + {{{\overset{\sim}{x}}_{1}\left( {n - i_{1}} \right)}{{\overset{\sim}{x}}_{1}\left( {n - i_{2}} \right)}{{\overset{\sim}{x}}_{1}^{*}\left( {n - i_{3}} \right)}{{\overset{\sim}{x}}_{2}^{*}\left( {n - i_{4}} \right)}{{\overset{\sim}{x}}_{2}\left( {n - i_{5}} \right)}} + {{{\overset{\sim}{x}}_{1}\left( {n - i_{1}} \right)}{{\overset{\sim}{x}}_{1}^{*}\left( {n - i_{2}} \right)}{{\overset{\sim}{x}}_{1}\left( {n - i_{3}} \right)}{{\overset{\sim}{x}}_{2}^{*}\left( {n - i_{4}} \right)}{{\overset{\sim}{x}}_{2}\left( {n - i_{5}} \right)}} + {{{\overset{\sim}{x}}_{1}^{*}\left( {n - i_{1}} \right)}{{\overset{\sim}{x}}_{1}\left( {n - i_{2}} \right)}{{\overset{\sim}{x}}_{2}^{*}\left( {n - i_{3}} \right)}{{\overset{\sim}{x}}_{2}^{*}\left( {n - i_{4}} \right)}{{\overset{\sim}{x}}_{2}\left( {n - i_{5}} \right)}}}} & \; \\{{{\overset{\sim}{x}}_{5,{d\; 2}}\left( {n,i_{1},i_{2},i_{3},i_{4},i_{5}} \right)} = {{{{\overset{\sim}{x}}_{1}\left( {n - i_{1}} \right)}{{\overset{\sim}{x}}_{2}\left( {n - i_{2}} \right)}{{\overset{\sim}{x}}_{2}\left( {n - i_{3}} \right)}{{\overset{\sim}{x}}_{2}^{*}\left( {n - i_{4}} \right)}{{\overset{\sim}{x}}_{2}^{*}\left( {n - i_{5}} \right)}} + {{{\overset{\sim}{x}}_{1}\left( {n - i_{1}} \right)}{{\overset{\sim}{x}}_{2}\left( {n - i_{2}} \right)}{{\overset{\sim}{x}}_{2}^{*}\left( {n - i_{3}} \right)}{{\overset{\sim}{x}}_{2}\left( {n - i_{4}} \right)}{{\overset{\sim}{x}}_{2}^{*}\left( {n - i_{5}} \right)}} + {{{\overset{\sim}{x}}_{1}\left( {n - i_{1}} \right)}{{\overset{\sim}{x}}_{2}^{*}\left( {n - i_{2}} \right)}{{\overset{\sim}{x}}_{2}\left( {n - i_{3}} \right)}{{\overset{\sim}{x}}_{2}\left( {n - i_{4}} \right)}{{\overset{\sim}{x}}_{2}^{*}\left( {n - i_{5}} \right)}} + {{{\overset{\sim}{x}}_{1}\left( {n - i_{1}} \right)}{{\overset{\sim}{x}}_{2}\left( {n - i_{2}} \right)}{{\overset{\sim}{x}}_{2}^{*}\left( {n - i_{3}} \right)}{{\overset{\sim}{x}}_{2}^{*}\left( {n - i_{4}} \right)}{{\overset{\sim}{x}}_{2}^{*}\left( {n - i_{5}} \right)}} + {{{\overset{\sim}{x}}_{1}\left( {n - i_{1}} \right)}{{\overset{\sim}{x}}_{2}^{*}\left( {n - i_{2}} \right)}{{\overset{\sim}{x}}_{2}\left( {n - i_{3}} \right)}{{\overset{\sim}{x}}_{2}^{*}\left( {n - i_{4}} \right)}{{\overset{\sim}{x}}_{2}\left( {n - i_{5}} \right)}} + {{{\overset{\sim}{x}}_{1}\left( {n - i_{1}} \right)}{{\overset{\sim}{x}}_{2}^{*}\left( {n - i_{2}} \right)}{{\overset{\sim}{x}}_{2}^{*}\left( {n - i_{3}} \right)}{{\overset{\sim}{x}}_{2}\left( {n - i_{4}} \right)}{{\overset{\sim}{x}}_{2}\left( {n - i_{5}} \right)}}}} & \;\end{matrix}$

In (28), M₁, M_(3,s), M_(3,d), M_(5,s), M_(5,d1) and M_(3,d2) denote thememory depth of the first, third, and fifth order Volterra seriesdistortion terms. The dual-band complex valued BBE Volterra Series in(28) includes only nonlinear distortion products of up to order 5.Expression of the dual band BBE Volterra model with higher nonlinearitycan be similarly derived. In addition, only odd powered terms areretained and even terms are discarded since they do not appear in thepass band. It is worth mentioning that the distortion terms {tilde over(x)}_(3,s)(n, i₁, i₂, i₃) and {tilde over (x)}_(5,s)(n, i₁, i₂, i₃, i₄,i₅) are linear combinations of three third and ten fifth orderdistortion products, respectively.

According to (28) and FIG. 5, the BBE Volterra is formed by twocategories of distortion terms, namely self-distortion terms 20 andinter-band band distortion terms 22. Each of these two families of termsrepresents a different dynamic distortion mechanism and therefore callfor different values for memory depth M₁, M_(3,s), M_(3,d), M_(5,s),M_(5,d1) and M_(5,d2) as shown in (28). Here, M₁, M_(3,s) and M_(5,s)represent the memory depth of the first, third and fifth order of theself-dynamic distortion terms, respectively, and M_(3,d), M_(5,d1) andM_(5,d2) represent the memory depth of the third and fifth orderinter-band dynamic distortion terms. Hence, the proposed modelformulation provides the capacity to separate memory depth values ineach distortion mechanism as opposed to the other approaches, such as2D-DPD, that use a global memory depth parameter M for all of thedistortion terms. This represents an additional degree of freedom forlimiting the implementation complexity of the multiband BBE.

An example of the dual band BBE Volterra model of (28) is given for NL=3and M₁=M_(3,s)=M_(3,d)=1 in (29).{tilde over (y)}(n)={tilde over (h)} _(ω) ₁ _(,1)(0){tilde over (x)}₁(n)+{tilde over (h)} _(ω) ₁ _(,1)(1){tilde over (x)} ₁(n−1) +3{tildeover (h)} _(ω) ₁ _(,3,s)(0,0,0){tilde over (x)} ₁(n){tilde over (x)}₁(n){tilde over (x)} ₁*(n)+{tilde over (h)} _(ω) ₁_(,3,s)(0,0,1)(2{tilde over (x)} ₁(n){tilde over (x)} ₁(n−1){tilde over(x)} ₁*(n)+{tilde over (x)} ₁(n){tilde over (x)} ₁(n){tilde over (x)}₁*(n−1))+{tilde over (h)} _(ω) ₁ _(,3,s)(0,1,1)(2{tilde over (x)}₁(n){tilde over (x)} ₁(n−1){tilde over (x)} ₁*(n−1)+{tilde over (x)}₁(n−1){tilde over (x)} ₁(n−1){tilde over (x)} ₁*(n))+3{tilde over (h)}_(ω) ₁ _(,3,s)(1,1,1){tilde over (x)} ₁(n−1){tilde over (x)}₁(n−1){tilde over (x)} ₁*(n−1)+2{tilde over (h)} _(ω) ₁_(,3,d)(0,0,0){tilde over (x)} ₁(n){tilde over (x)} ₂(n){tilde over (x)}₂*(n)+{tilde over (h)} _(ω) ₁ _(,3,d)(0,0,1)({tilde over (x)} ₁(n){tildeover (x)} ₂(n){tilde over (x)} ₂*(n−1)+{tilde over (x)}₁(n){tilde over(x)} ₂*(n){tilde over (x)} ₂(n−1))+2{tilde over (h)} _(ω) ₁_(,3,d)(0,1,1){tilde over (x)} ₁(n){tilde over (x)} ₂(n−1){tilde over(x)} ₂*(n−1)+2{tilde over (h)} _(ω) ₁ _(,3,d)(1,0,0){tilde over (x)}₁(n−1){tilde over (x)} ₂(n){tilde over (x)} ₂*(n)+{tilde over (h)} _(ω)₁ _(,3,d)(1,0,1)({tilde over (x)} ₁(n−1){tilde over (x)} ₂(n){tilde over(x)} ₂*(n−1)+{tilde over (x)}₁(n−1){tilde over (x)} ₂*(n){tilde over(x)} ₂(n−1))+2{tilde over (h)} _(ω) ₁ _(,3,d)(1,1,1){tilde over (x)}₁(n−1){tilde over (x)} ₂(n−1){tilde over (x)} ₂*(n−1)   (29)

A close examination of (28) reveals a number of important attributes ofthe dual band BBE Volterra model: Inclusion of all the possibledistortion terms attributed to the static and dynamic nonlinear behaviorof the PA. These involve either only the envelope of the first bandsignal, e.g. {tilde over (x)}₁*(n){tilde over (x)}₁ ²(n), and {tildeover (x)}₁*(n){tilde over (x)}₁(n){tilde over (x)}₁(n−1), or result fromthe mixing between the two bands' envelopes, e.g. {tilde over(x)}₁(n){tilde over (x)}₂*(n){tilde over (x)}₂(n), and {tilde over(x)}₁(n){tilde over (x)}₂*(n−1){tilde over (x)}₂(n). A large number ofthe distortion terms included in (28) were not incorporated in the2D-DPD model (i.e., {tilde over (x)}₂*(n){tilde over (x)}₂(n){tilde over(x)}₁(n−1) and {tilde over (x)}₂*(n−1){tilde over (x)}₂(n){tilde over(x)}₁(n−1)).

While the dual band BBE Volterra model described herein includes largernumber of distortion products than other models, according to (28) theseproducts are grouped into different sets. Each set forms a distortionterm, e. g. the distortion term {tilde over (x)}_(3,s)(n, i₁, i₂, i₃)represents the grouping of the following distortion products {tilde over(x)}₁(n−i₁){tilde over (x)}₁(n−i₂){tilde over (x)}₁*(n−i₃), {tilde over(x)}₁(n−i₁){tilde over (x)}₁*(n−i₂){tilde over (x)}₁(n−i₃),{tilde over(x)}₁*(n−i₁){tilde over (x)}₁(n−i₂){tilde over (x)}₁(n−i₃). Thedistortion products that belong to a given set share the same kernel.For example, for every possible triplet (i₁, i₂, i₃) ε {0 . . . M}³,three 3^(rd) order distortion products are combined to form {tilde over(x)}_(3,s)(n, i₁, i₂, i₃) and consequently share one kernel {tilde over(h)}_(ω) ₁ _(,3,s)(i₁, i₂, i₃) in (28). Similarly, for every possiblequintuplet (i_(i), i₂, i₃, i₄, i₅) ε {0 . . . M}⁵, ten 5^(th) orderdistortion products are combined to form {tilde over (x)}_(5,s)(n, i₁,i₂, i₃, i₄, i₅) and share one kernel {tilde over (h)}_(ω) ₁ _(,5,s)(i₁,i₂, i₃, i₄, i₅) in (28). Hence, despite the fact that the modelsdescribed herein involve more distortion terms, the models usecomparable numbers of kernels compared to the 2D-DPD scheme.

The expression of the dual band BBE Volterra model of (28) preserves thelinearity property with respect to its coefficients. Hence, the leastsquare error (LSE) estimator can be applied to identify the kernels in(28) for a given RF PA. Equation (30) details the expression used tocompute the LSE solution of (28):A·h=Y   (30)where A denotes the distortion products matrix, h is the kernels' vectorto be estimated and Y is the vector formed by the output signal sample.Each of these variables (A, h and Y) is defined in (31) where Lrepresents the data stream size:

$A = \begin{pmatrix}{{\overset{\sim}{x}}_{1,s}\left( {M_{1} + 1} \right)} & \cdots & {{\overset{\sim}{x}}_{1,s}(1)} & {{\overset{\sim}{x}}_{3,s}\left( {{M_{3,s} + 1},0,0,0} \right)} & \cdots & {{\overset{\sim}{x}}_{3,s}\left( {1,M_{3,s},M_{3,5},M_{3,s}} \right)} & {{\overset{\sim}{x}}_{3,d}\left( {{M_{3,d}\_ 1},0,0,0,0,0} \right)} & \cdots & \cdots \\\vdots & \; & \vdots & \vdots & \; & \vdots & \vdots & \; & \vdots \\\vdots & \ddots & \vdots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\{{\overset{\sim}{x}}_{1,s}(L)} & \cdots & {{\overset{\sim}{x}}_{1,s}\left( {L - M_{1}} \right)} & {{\overset{\sim}{x}}_{3,s}\left( {L,0,0,0} \right)} & \cdots & {{\overset{\sim}{x}}_{3,s}\left( {{L - M_{3,s}},M_{3,s},M_{3,5},M_{3,5}} \right)} & {{\overset{\sim}{x}}_{3,d}\left( {L,0,0,0,0,0} \right)} & \cdots & \cdots\end{pmatrix}$ $h = \begin{matrix}\left( h_{0} \right. & \cdots & h_{M_{1}} & h_{0,0,0} & {\left. \cdots \right)^{T},}\end{matrix}$ $Y = {\begin{matrix}\left( {{\overset{\sim}{y}}_{1,0}\left( {M_{1} + 1} \right)} \right. & \cdots & \left. {{\overset{\sim}{y}}_{1,0}(L)} \right)^{T}\end{matrix}\mspace{1024mu}(31)}$The LSE solution to equation (30) is computed using:ĥ=(A ^(T) ·A)⁻¹ ·A ^(T) ·Y  (32)where ĥ is the estimate of h.

To summarize the approach, a discrete base band equivalent, BBE,Volterra series is generated based on a received multi-band signal. Theseries has distortion products grouped according to determined sharedkernels. The shared kernels are determined based on a transformation ofa real-valued continuous-time pass band Volterra series without pruningof kernels. The transformation includes transforming the real-valuedcontinuous time pass band Volterra series to a multi-frequencycomplex-valued envelope series. The multi-frequency complex-valuedenvelope signal is then transformed to a continuous-time pass band-onlyseries. The continuous-time pass band-only signal is transformed to acontinuous-time baseband equivalent series. The continuous-time basebandequivalent signal is discretized to produce the discrete base bandequivalent Volterra series. Shared kernels of the discrete base bandequivalent Volterra series are identified, where each shared kernel hasdistortion products in common with another shared kernel.

To assess the performance of the model described above, the formulationwas used to model and linearize a high power dual-band RF PA. The deviceunder test was a broadband 45 W single ended GaN PA driven with adual-band multi-standard signal. Three test scenarios are defined:

-   -   Case 1: 20MHz (1001) WCDMA and 20 MHz LTE signals centered @2.1        GHz and 2.2 GHz respectively. The PAPR of the dual-band signal        is equal to 10.3 dB.    -   Case 2: 20 MHz (1001) WCDMA and 20 MHz LTE signals centered @2.1        GHz and 2.4 GHz, respectively. The PAPR of the dual-band signal        is equal to 10.3 dB.    -   Case 3: 20 MHz 4 C WCDMA and 20 MHz LTE signals centered @2.1        GHz and 2.8 GHz respectively. The PAPR of the dual-band signal        is equal to 10.1 dB.

The proposed dual-band BBE Volterra and 2D-DPD models were each used tolinearize a DUT PA. The training of the two models was conducted usingsamples of the PA output signals in each band sampled at 100MSPS. Thenonlinearity order and memory depth of each model were individually setto achieve the best performance versus complexity trade-off in eachcase. For the first and third cases, the 2D-DPD and dual-band BBEVolterra parameters were set to (NL=7 and M=3) and (NL=7, M₁=2,M_(3,s)=M_(3,d)=1, M_(5,s)=M_(5,d1)=M_(5,d2)=0, M_(7,s)=0),respectively. However, in Case 2, (NL=9 and M=3) and (NL=9, M₁=2,M_(3,s)=M_(3,d)=1, M_(5,s)=M_(5,d1) =M_(5,d2)=0, M₇=0, M₉=0) were foundto be adequate for the 2D-DPD and the dual-band BBE Volterra,respectively.

The linearization results for the three test scenarios are shown inFIGS. 6-11 and summaries of the corresponding performances are given inTable I-III. In each of FIGS. 6-11 the results are given for no DPD, for2D-DPD and for the linearization method described herein. In all testscenarios, the DIDO BBE Volterra model successfully linearized the PAwith significantly lower complexity and slightly better performance thanthe 2D-DPD model. A reduction of the spectrum regrowth of about 20 dBand an ACPR of about 50 dBc were achieved by the dual-band BBE Volterrain all test scenarios.

TABLE I Dual band standard linearization results: Case 1 Without DPDWith Volterra DPD With 2D-DPD Band 1 Band 2 Band 1 Band 2 Band 1 Band 2@ 2.1 @ 2.2 @ 2.1 @ 2.2 @ 2.1 @ 2.2 GHz GHz GHz GHz GHz GHz Number of 00 20 20 84 84 coefficients NMSE −19 −18 −38 −37 −37 −36 (dB) ACLR −35−27 −51 −48 −50 −46 (dBc)

TABLE II Dual band standard linearization results: Case 2 Without DPDWith Volterra DPD With 2D-DPD Band 1 Band 2 Band 1 Band 2 Band 1 Band 2@ 2.1 @ 2.4 @ 2.1 @ 2.4 @ 2.1 @ 2.4 GHz GHz GHz GHz GHz GHz Number of 00 25 25 135 135 coefficients NMSE −18 −17 −38 −36 −36 −36 (dB) ACLR −35−25 −51 −47 −48 −45 (dBc)

TABLE III Dual band standard linearization results: Case 3 Without DPDWith Volterra DPD With 2D-DPD Band 1 Band 2 Band 1 Band 2 Band 1 Band 2@ 2.1 @ 2.8 @ 2.1 @ 2.8 @ 2.1 @ 2.8 GHz GHz GHz GHz GHz GHz Number of 00 20 20 84 84 coefficients NMSE −19 −17 −38 −35 −36 −35 (dB) ACLR −32−27 −48 −47 −47 −47 (dBc)

FIG. 12 is a block diagram of a power amplification system 24 having adigital pre-distorter modelling unit 26 implementing the dual band BBEVolterra model presented herein. Note that although FIG. 12 shows onlytwo bands, the invention is not limited to two bands but rather, can beimplemented for more than two bands according to the steps describedabove. The power amplification system 24 includes digital pre-distorters28 a and 28 b, referred to collectively as DPDs 28. The DPDs 28 receiveinput from the pre-distorter modelling unit 26, and pre-distort theinput signals {tilde over (x)}₁ and {tilde over (x)}₂ to producepre-distorted signals {tilde over (x)}_(1p) and {tilde over (x)}_(2p).Each pre-distorted signal is input to a digital modulator 30 to impressthe baseband signal onto a respective carrier, converted to analog by adigital (A/D) to analog converter 32, low pass filtered by a filter 34,and mixed to radio frequency (RF) by a mixer 36 to prepare the signalfor amplification by an RF PA amplifier. Accordingly, the RF signals inthe two paths are summed by an adder 38 and input to a power amplifier40. A transmitter observation receiver 42 samples the output of thepower amplifier 40 in each band and produces output signals y_(w1) andy_(w2) These output signals are used by the DPD modeling unit 26 toderive the kernel vector h according to equation (30). The DPD modellingunit 26 calculates a discrete baseband equivalent Volterra series havingdistortion products grouped according to determined shared kernels,where the shared kernels are based on a transformation of a real-valuedcontinuous-time pass band Volterra series without pruning of kernels.

FIG. 13 is a more detailed view of the DPD modelling unit 26, whichincludes a memory module 44 in communication with a processor 46. TheDPD modelling unit 26 receives the input signals {tilde over (x)}₁ and{tilde over (x)}₂ and output signals {tilde over (y)}_(ω1) and {tildeover (ω)}_(ω2) from the transmitter observation receiver 42 and derivesthe modelling vector h according to equation (32). The modelling vectorh. is input to the DPDs 28 to pre-distort the input signals {tilde over(x)}₁ and {tilde over (x)}₂ to produce pre-distorted signals {tilde over(x)}_(1p) and {tilde over (x)}_(2p). The processor 46 includes agrouping module 50, a shared kernel determiner 52 and a series termcomputer 54. The grouping module is configured to group distortionproducts of the series according to determined shared kernels. Theshared kernel determiner is configured to determine the shared kernelsbased on a transformation of a real-valued continuous-time pass bandVolterra series without pruning of kernels. The series term calculatoris configured to calculate the terms of the discrete base bandequivalent Volterra series, the terms being the distortion productsmultiplied by their respective shared kernels. The memory module 44 isconfigured to store terms of the discrete base band equivalent (BBE)Volterra series 48, generated by the processor 46.

FIG. 14 is a flowchart of a process for modelling a power amplifier 40fed by a multi-band input signal. A multi-band signal is received by adigital pre-distorter 28 (block S100). A discrete BBE Volterra series isgenerated by the DPD modelling unit 26 based on the received multi-bandinput signal (block S102). The series has distortion products that aregrouped by the grouping module 50 according to determined shared kernelsdetermined by shared kernel determiner 52. The shared kernels aredetermined based on a transformation of a real-valued continuous-timepass band Volterra series without pruning of kernels.

FIG. 15 is a flowchart of a process for transforming a real-valuedcontinuous time pass band Volterra series to a discrete base bandequivalent Volterra series in which shared kernels are identified as setout in block S102 of FIG. 14. The real-valued continuous time pass bandVolterra series is transformed to a multi-frequency complex-valuedenvelope series (block S104). The multi-frequency complex-valuedenvelope signal is transformed to a continuous-time pass band-onlyseries (block S106). The continuous-time pass band-only signal istransformed to a continuous-time baseband equivalent series (blockS108). The continuous-time baseband equivalent signal is discretized toproduce the discrete base band equivalent Volterra series (block S110).Shared kernels of the discrete base band equivalent Volterra series areidentified, where a shared kernel has distortion products in common withanother shared kernel (block S112).

FIG. 16 is a flowchart of a process of transforming the continuous-timepass band-only signal to a continuous-time baseband equivalent signal asshown in block S108 of FIG. 15. The continuous-time pass band-onlyseries is expressed in convolution form (block S114). Then, the Laplacetransform is applied to the convolution form to produce a Laplace domainexpression (block S116). A number of terms in the Laplace domainexpression may be reduced based on symmetry (block S118). The Laplacedomain expression is frequency-shifted to baseband to produce a basebandequivalent expression in the Laplace domain (block S120). An inverseLaplace transform is applied to the baseband equivalent expression toproduce the continuous-time baseband equivalent series (block S122).

FIG. 17 is a flowchart of a process of discretizing the continuous-timebaseband equivalent series to produce the discrete base band equivalentVolterra series, as shown in block S110 of FIG. 15. The process includestruncating the continuous-time baseband equivalent series to a finitenon-linearity order (block S124). The process also includes expressingthe truncated series as summations of non-linear distortion terms, withupper limits of the summations being memory depths assigned to eachorder of the non-linear distortion terms (block S126).

Thus, a dual band BBE Volterra series-based behavioral model has beendescribed herein to mimic and linearize the dynamic nonlinear behaviorof a concurrently driven dual-band amplifier. Starting with areal-valued, continuous-time, pass band Volterra series and using anumber of signal and system transformations, a low complexitycomplex-valued, and discrete BBE Volterra formulation was derived. Whilethe formulation presented herein includes all possible distortion terms,it involved fewer kernels than its 2D-DPD counterpart. The model issuccessfully applied to digitally predistort and linearize a dual-band45 Watt class AB GaN PA driven with different dual-band dual-standardtest signals. For each band, the model used less than 25 coefficients toreduce the ACLR by up to 25 dB.

It will be appreciated by persons skilled in the art that the presentinvention is not limited to what has been particularly shown anddescribed herein above. In addition, unless mention was made above tothe contrary, it should be noted that all of the accompanying drawingsare not to scale. A variety of modifications and variations are possiblein light of the above teachings without departing from the scope andspirit of the invention, which is limited only by the following claims

What is claimed is:
 1. A method of modelling a power amplifier fed by amulti-band signal input, the method comprising: receiving a multi-bandsignal; generating a discrete base band equivalent, BBE, Volterra seriesbased on the received multi-band signal, the discrete BBE Volterraseries having distortion products grouped according to determined sharedkernels, the discrete BBE Volterra series being used to model the poweramplifier; and applying the discrete BBE Volterra series to pre-distortthe multi-band signal and to linearize the power amplifier; and theshared kernels being determined based on a transformation of areal-valued continuous-time pass band Volterra series without pruning ofkernels.
 2. The method of claim 1, wherein the shared kernels aredetermined based on the transformation of the real-valuedcontinuous-time pass band Volterra series by: transforming thereal-valued continuous time pass band Volterra series to amulti-frequency complex-valued envelope series; transforming themulti-frequency complex-valued envelope signal to a continuous-time passband-only series; transforming the continuous-time pass band-only signalto a continuous-time baseband equivalent series; discretizing thecontinuous-time baseband equivalent signal to produce the discrete baseband equivalent Volterra series; and identifying the shared kernels,each shared kernel having distortion products in common with anothershared kernel.
 3. The method of claim 2, wherein transforming thecontinuous-time pass band-only signal to a continuous-time basebandequivalent signal includes: expressing the continuous-time passband-only series in convolution form; applying a Laplace transform tothe convolution form to produce a Laplace domain expression; frequencyshifting the Laplace domain expression to baseband to produce a basebandequivalent expression in the Laplace domain; and applying an inverseLaplace transform to the baseband equivalent expression to produce thecontinuous-time baseband equivalent series.
 4. The method of claim 3,wherein a number of terms in the Laplace domain expression are reducedvia symmetry.
 5. The method of claim 3, further comprising groupingterms of the Laplace domain expression based on frequency intervalswhere distortion terms are not zero.
 6. The method of claim 2, whereindiscretizing the continuous-time baseband equivalent series to producethe discrete base band equivalent Volterra series includes: truncatingthe continuous-time baseband equivalent series to a finite non-linearityorder; and expressing the truncated series as summations of non-lineardistortion terms, with upper limits of the summations being memorydepths assigned to each order of the non-linear distortion terms.
 7. Themethod of claim 6, wherein a distortion term is a group of distortionproducts multiplied by a shared kernel.
 8. A digital pre-distorter (DPD)system including a power amplifier receiving a multi-band input, thesystem comprising: a Volterra series DPD modelling unit, the DPDmodelling unit configured to: calculate a discrete base band equivalent,BBE, Volterra series, the discrete BBE Volterra series having distortionproducts grouped according to determined shared kernels; applying thediscrete BBE Volterra series to pre-distort the multi-band signal and tolinearize the power amplifier; and the shared kernels being determinedbased on a transformation of a real-valued continuous-time pass bandVolterra series without pruning of kernels.
 9. The DPD system of claim8, wherein the power amplifier is configured to produce an output inresponse to the multi-band input, the output of the power amplifierprovided to the Volterra series DPD modelling unit to enable theVolterra series DPD modeling unit to compute the shared kernels based onthe output of the power amplifier.
 10. The DPD system of claim 9,further comprising a transmitter observation receiver configured tosample the output of the power amplifier and provide the sampled outputto the Volterra series DPD modelling unit.
 11. The DPD system of claim8, wherein the distortion products and their associated kernels aredetermined by: transforming the real-valued continuous time pass bandVolterra series to a multi-frequency complex-valued envelope series;transforming the multi-frequency complex-valued envelope signal to acontinuous-time pass band-only series; transforming the continuous-timepass band-only signal to a continuous-time baseband equivalent series;discretizing the continuous-time baseband equivalent signal to producethe discrete base band equivalent Volterra series; and identifying theshared kernels, each shared kernel having distortion products in commonwith another shared kernel.
 12. The DPD system of claim 11, whereintransforming the continuous-time pass band-only signal to acontinuous-time baseband equivalent signal includes: expressing thecontinuous-time pass band-only series in convolution form; applying aLaplace transform to the convolution form to produce a Laplace domainexpression; frequency shifting the Laplace domain expression to basebandto produce a baseband equivalent expression in the Laplace domain; andapplying an inverse Laplace transform to the baseband equivalentexpression to produce the continuous-time baseband equivalent series.13. The DPD system of claim 12, wherein a number of terms in the Laplacedomain expression are reduced via symmetry.
 14. The DPD system of claim12, wherein transforming the continuous-time pass band-only signal to acontinuous-time baseband equivalent signal further comprises groupingterms of the Laplace domain expression based on frequency intervalswhere distortion terms are not zero.
 15. The DPD system of claim 11,wherein discretizing the continuous-time baseband equivalent series toproduce the discrete base band equivalent Volterra series includes:truncating the continuous-time baseband equivalent series to a finitenon-linearity order; and expressing the truncated series as summationsof non-linear distortion terms, with upper limits of the summationsbeing memory depths assigned to each order of the non-linear distortionterms.
 16. The DPD system of claim 15, wherein a distortion term is agroup of distortion products multiplied by a shared kernel.
 17. AVolterra series digital pre-distorter, DPD, modelling unit for modelingdistortion of a power amplifier receiving a multi-band input, themodelling unit comprising: a memory module, the memory module configuredto store terms of a discrete base band equivalent, BBE, Volterra series;a grouping module, the grouping module configured to group distortionproducts of the discrete BBE Volterra series according to determinedshared kernels; a shared kernel determiner, the shared kernel determinerconfigured to determine the shared kernels based on a transformation ofa real-valued continuous-time pass band Volterra series without pruningof kernels; and a series term calculator, the series term calculatorconfigured to calculate the terms of the discrete base band equivalentVolterra series, the terms being the distortion products multiplied bytheir respective shared kernels; the modelling unit being furtherconfigured to send the discrete BBE Volterra series to a digitalpre-distorter to pre-distort the multi-band signal and to linearize thepower amplifier.
 18. The Volterra series DPD modelling unit of claim 17,wherein the BBE Volterra series terms are based on the multi-band input.19. The Volterra series DPD modelling unit of claim 18, wherein themulti-band input is a dual band input.
 20. The Volterra series DPDmodelling unit of claim 17, wherein the shared kernel determiner isfurther configured to determine the shared kernels via a least squaresestimate based on the multi-band input and an output of the poweramplifier.
 21. The Volterra series DPD modelling unit of claim 17,wherein the kernels and distortion products are derived from thereal-valued continuous-time pass band Volterra series by: transformingthe real-valued continuous time pass band Volterra series to amulti-frequency complex-valued envelope series; transforming themulti-frequency complex-valued envelope signal to a continuous-time passband-only series; transforming the continuous-time pass band-only signalto a continuous-time baseband equivalent series; discretizing thecontinuous-time baseband equivalent signal to produce the discrete baseband equivalent Volterra series; and identifying the shared kernels,each shared kernel having distortion products in common.